Theorem vtoclegftOLD 2357

Description: Implicit substitution of a class for a set variable. (Closed theorem version of vtoclef 2358.)

Assertion

Ref	Expression
vtoclegftOLD	|- ((A e. B /\ A.x(ph -> A.xph) /\ A.x(x = A -> ph)) -> ph)

Distinct variable group:   x,A

Proof of Theorem vtoclegftOLD

Step	Hyp	Ref	Expression
1		19.23t 1474	. . . 4 |- (A.x(ph -> A.xph) -> (A.x(x = A -> ph) <-> (E.x x = A -> ph)))
2	1	adantl 424	. . 3 |- ((A e. B /\ A.x(ph -> A.xph)) -> (A.x(x = A -> ph) <-> (E.x x = A -> ph)))
3		elex 2302	. . . . 5 |- (A e. B -> E.x x = A)
4		pm2.27 76	. . . . 5 |- (E.x x = A -> ((E.x x = A -> ph) -> ph))
5	3, 4	syl 12	. . . 4 |- (A e. B -> ((E.x x = A -> ph) -> ph))
6	5	adantr 425	. . 3 |- ((A e. B /\ A.x(ph -> A.xph)) -> ((E.x x = A -> ph) -> ph))
7	2, 6	sylbid 220	. 2 |- ((A e. B /\ A.x(ph -> A.xph)) -> (A.x(x = A -> ph) -> ph))
8	7	3impia 1064	1 |- ((A e. B /\ A.x(ph -> A.xph) /\ A.x(x = A -> ph)) -> ph)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294

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