Theorem vtoclegft 2356

Description: Implicit substitution of a class for a set variable. (Closed theorem version of vtoclef 2358.) (The proof was shortened by Andrew Salmon, 8-Jun-2011.)

Assertion

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

Distinct variable group:   x,A

Proof of Theorem vtoclegft

Step	Hyp	Ref	Expression
1		elex 2302	. . 3 |- (A e. B -> E.x x = A)
2		pm2.27 76	. . . . 5 |- (E.x x = A -> ((E.x x = A -> E.xph) -> E.xph))
3		19.9t 1382	. . . . 5 |- (A.x(ph -> A.xph) -> (E.xph -> ph))
4	2, 3	syl9 71	. . . 4 |- (E.x x = A -> (A.x(ph -> A.xph) -> ((E.x x = A -> E.xph) -> ph)))
5		exim 1386	. . . 4 |- (A.x(x = A -> ph) -> (E.x x = A -> E.xph))
6	4, 5	syl7 26	. . 3 |- (E.x x = A -> (A.x(ph -> A.xph) -> (A.x(x = A -> ph) -> ph)))
7	1, 6	syl 12	. 2 |- (A e. B -> (A.x(ph -> A.xph) -> (A.x(x = A -> ph) -> ph)))
8	7	3imp 1061	1 |- ((A e. B /\ A.x(ph -> A.xph) /\ A.x(x = A -> ph)) -> ph)

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

This theorem is referenced by:  elabgtOLD 2401

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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