Theorem vtoclegft 1847

Description: Implicit substitution of a class for a set variable. (Closed theorem version of vtoclef 1848.)

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		19.23t 1112	. . . 4 |- (A.x(ph -> A.xph) -> (A.x(x = A -> ph) <-> (E.x x = A -> ph)))
2	1	adantl 388	. . 3 |- ((A e. B /\ A.x(ph -> A.xph)) -> (A.x(x = A -> ph) <-> (E.x x = A -> ph)))
3		elex 1810	. . . . 5 |- (A e. B -> E.x x = A)
4		pm2.27 62	. . . . 5 |- (E.x x = A -> ((E.x x = A -> ph) -> ph))
5	3, 4	syl 10	. . . 4 |- (A e. B -> ((E.x x = A -> ph) -> ph))
6	5	adantr 389	. . 3 |- ((A e. B /\ A.x(ph -> A.xph)) -> ((E.x x = A -> ph) -> ph))
7	2, 6	sylbid 203	. 2 |- ((A e. B /\ A.x(ph -> A.xph)) -> (A.x(x = A -> ph) -> ph))
8	7	3impia 828	1 |- ((A e. B /\ A.x(ph -> A.xph) /\ A.x(x = A -> ph)) -> ph)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 773  A.wal 951   = wceq 953   e. wcel 955  E.wex 977

This theorem is referenced by:  elabgt 1886

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803

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