Theorem hbsb4t 1249

Description: A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 1248).

Assertion

Ref	Expression
hbsb4t	|- (A.xA.z(ph -> A.zph) -> (-. A.z z = y -> ([y / x]ph -> A.z[y / x]ph)))

Proof of Theorem hbsb4t

Step	Hyp	Ref	Expression
1		ax-4 973	. . . . . 6 |- (A.zph -> ph)
2	1	biantru 724	. . . . 5 |- ((ph -> A.zph) <-> ((ph -> A.zph) /\ (A.zph -> ph)))
3		dfbi2 514	. . . . 5 |- ((ph <-> A.zph) <-> ((ph -> A.zph) /\ (A.zph -> ph)))
4	2, 3	bitr4 176	. . . 4 |- ((ph -> A.zph) <-> (ph <-> A.zph))
5	4	2albii 1000	. . 3 |- (A.xA.z(ph -> A.zph) <-> A.xA.z(ph <-> A.zph))
6		a4sbbi 1245	. . . . . 6 |- (A.x(ph <-> A.zph) -> ([y / x]ph <-> [y / x]A.zph))
7	6	a4s 984	. . . . 5 |- (A.zA.x(ph <-> A.zph) -> ([y / x]ph <-> [y / x]A.zph))
8		hba1 1003	. . . . . 6 |- (A.zA.x(ph <-> A.zph) -> A.zA.zA.x(ph <-> A.zph))
9	8, 7	albid 1104	. . . . 5 |- (A.zA.x(ph <-> A.zph) -> (A.z[y / x]ph <-> A.z[y / x]A.zph))
10	7, 9	imbi12d 626	. . . 4 |- (A.zA.x(ph <-> A.zph) -> (([y / x]ph -> A.z[y / x]ph) <-> ([y / x]A.zph -> A.z[y / x]A.zph)))
11	10	a7s 991	. . 3 |- (A.xA.z(ph <-> A.zph) -> (([y / x]ph -> A.z[y / x]ph) <-> ([y / x]A.zph -> A.z[y / x]A.zph)))
12	5, 11	sylbi 199	. 2 |- (A.xA.z(ph -> A.zph) -> (([y / x]ph -> A.z[y / x]ph) <-> ([y / x]A.zph -> A.z[y / x]A.zph)))
13		hba1 1003	. . 3 |- (A.zph -> A.zA.zph)
14	13	hbsb4 1248	. 2 |- (-. A.z z = y -> ([y / x]A.zph -> A.z[y / x]A.zph))
15	12, 14	syl5bir 210	1 |- (A.xA.z(ph -> A.zph) -> (-. A.z z = y -> ([y / x]ph -> A.z[y / x]ph)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956  [wsbc 1170

This theorem is referenced by:  dvelimdf 1251  hbabd 1468

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-11o 1218

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172

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