Theorem sbcgf 1996

Description: Substitution for a variable not free in a wff does not affect it.

Hypothesis

Ref	Expression
sbcgf.1	|- (ph -> A.xph)

Assertion

Ref	Expression
sbcgf	|- (A e. B -> ([A / x]ph <-> ph))

Proof of Theorem sbcgf

Step	Hyp	Ref	Expression
1		sbccog 1959	. 2 |- (A e. B -> ([A / y][y / x]ph <-> [A / x]ph))
2		sbcgf.1	. . . . 5 |- (ph -> A.xph)
3	2	sbf 1190	. . . 4 |- ([y / x]ph <-> ph)
4	3	sbcbii 1988	. . 3 |- (A e. B -> ([A / y][y / x]ph <-> [A / y]ph))
5		sbc5g 1961	. . 3 |- (A e. B -> ([A / y]ph <-> E.y(y = A /\ ph)))
6		elex 1826	. . . . 5 |- (A e. B -> E.y y = A)
7	6	biantrurd 731	. . . 4 |- (A e. B -> (ph <-> (E.y y = A /\ ph)))
8		19.41v 1309	. . . 4 |- (E.y(y = A /\ ph) <-> (E.y y = A /\ ph))
9	7, 8	syl6rbbr 542	. . 3 |- (A e. B -> (E.y(y = A /\ ph) <-> ph))
10	4, 5, 9	3bitrd 547	. 2 |- (A e. B -> ([A / y][y / x]ph <-> ph))
11	1, 10	bitr3d 533	1 |- (A e. B -> ([A / x]ph <-> ph))

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

This theorem is referenced by:  sbc19.21g 1997  sbcabel 2006  csbconstgf 2021  sbcel12g 2022  intab 2574  csbopabg 2693  dfoprab5 4131  foprab2 4135  fsumcnlem 8015

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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