Theorem sbcabel 2006

Description: Interchange class substitution and class abstraction.

Hypothesis

Ref	Expression
sbcabel.1	|- (z e. B -> A.x z e. B)

Assertion

Ref	Expression
sbcabel	|- (A e. C -> ([A / x]{y | ph} e. B <-> {y | [A / x]ph} e. B))

Distinct variable groups:   y,A   z,B   x,y   x,z

Proof of Theorem sbcabel

Step	Hyp	Ref	Expression
1		elisset 1824	. 2 |- (A e. C -> A e. V)
2		df-clel 1478	. . . . 5 |- ({y | ph} e. B <-> E.w(w = {y | ph} /\ w e. B))
3	2	sbcbii 1988	. . . 4 |- (A e. V -> ([A / x]{y | ph} e. B <-> [A / x]E.w(w = {y | ph} /\ w e. B)))
4		sbcexg 1985	. . . 4 |- (A e. V -> ([A / x]E.w(w = {y | ph} /\ w e. B) <-> E.w[A / x](w = {y | ph} /\ w e. B)))
5		sbcang 1981	. . . . . 6 |- (A e. V -> ([A / x](w = {y | ph} /\ w e. B) <-> ([A / x]w = {y | ph} /\ [A / x]w e. B)))
6		abeq2 1575	. . . . . . . . . 10 |- (w = {y | ph} <-> A.y(y e. w <-> ph))
7	6	sbcbii 1988	. . . . . . . . 9 |- (A e. V -> ([A / x]w = {y | ph} <-> [A / x]A.y(y e. w <-> ph)))
8		sbcalg 1984	. . . . . . . . 9 |- (A e. V -> ([A / x]A.y(y e. w <-> ph) <-> A.y[A / x](y e. w <-> ph)))
9		sbcbidig 1983	. . . . . . . . . . 11 |- (A e. V -> ([A / x](y e. w <-> ph) <-> ([A / x]y e. w <-> [A / x]ph)))
10		ax-17 975	. . . . . . . . . . . . 13 |- (y e. w -> A.x y e. w)
11	10	sbcgf 1996	. . . . . . . . . . . 12 |- (A e. V -> ([A / x]y e. w <-> y e. w))
12	11	bibi1d 622	. . . . . . . . . . 11 |- (A e. V -> (([A / x]y e. w <-> [A / x]ph) <-> (y e. w <-> [A / x]ph)))
13	9, 12	bitrd 531	. . . . . . . . . 10 |- (A e. V -> ([A / x](y e. w <-> ph) <-> (y e. w <-> [A / x]ph)))
14	13	albidv 1282	. . . . . . . . 9 |- (A e. V -> (A.y[A / x](y e. w <-> ph) <-> A.y(y e. w <-> [A / x]ph)))
15	7, 8, 14	3bitrd 547	. . . . . . . 8 |- (A e. V -> ([A / x]w = {y | ph} <-> A.y(y e. w <-> [A / x]ph)))
16		abeq2 1575	. . . . . . . 8 |- (w = {y | [A / x]ph} <-> A.y(y e. w <-> [A / x]ph))
17	15, 16	syl6bbr 541	. . . . . . 7 |- (A e. V -> ([A / x]w = {y | ph} <-> w = {y | [A / x]ph}))
18		ax-17 975	. . . . . . . . 9 |- (z e. w -> A.x z e. w)
19		sbcabel.1	. . . . . . . . 9 |- (z e. B -> A.x z e. B)
20	18, 19	hbel 1573	. . . . . . . 8 |- (w e. B -> A.x w e. B)
21	20	sbcgf 1996	. . . . . . 7 |- (A e. V -> ([A / x]w e. B <-> w e. B))
22	17, 21	anbi12d 631	. . . . . 6 |- (A e. V -> (([A / x]w = {y | ph} /\ [A / x]w e. B) <-> (w = {y | [A / x]ph} /\ w e. B)))
23	5, 22	bitrd 531	. . . . 5 |- (A e. V -> ([A / x](w = {y | ph} /\ w e. B) <-> (w = {y | [A / x]ph} /\ w e. B)))
24	23	exbidv 1283	. . . 4 |- (A e. V -> (E.w[A / x](w = {y | ph} /\ w e. B) <-> E.w(w = {y | [A / x]ph} /\ w e. B)))
25	3, 4, 24	3bitrd 547	. . 3 |- (A e. V -> ([A / x]{y | ph} e. B <-> E.w(w = {y | [A / x]ph} /\ w e. B)))
26		df-clel 1478	. . 3 |- ({y | [A / x]ph} e. B <-> E.w(w = {y | [A / x]ph} /\ w e. B))
27	25, 26	syl6bbr 541	. 2 |- (A e. V -> ([A / x]{y | ph} e. B <-> {y | [A / x]ph} e. B))
28	1, 27	syl 10	1 |- (A e. C -> ([A / x]{y | ph} e. B <-> {y | [A / x]ph} e. B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 958   = wceq 960   e. wcel 962  E.wex 984  [wsbc 1174  {cab 1469  Vcvv 1818

This theorem is referenced by:  csbexg 2019

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-9 969  ax-10 970  ax-11 971  ax-12 972  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1127  ax-10o 1144  ax-16 1214  ax-11o 1222  ax-ext 1464

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 985  df-sb 1176  df-clab 1470  df-cleq 1475  df-clel 1478  df-v 1819  df-sbc 1949

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