Theorem vtoclegft 1853

Description: Implicit substitution of a class for a set variable. (Closed theorem version of vtoclef 1854.)

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 1115	. . . 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 1816	. . . . 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 829	1 |- ((A e. B /\ A.x(ph -> A.xph) /\ A.x(x = A -> ph)) -> ph)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 774  A.wal 953   = wceq 955   e. wcel 957  E.wex 979

This theorem is referenced by:  elabgt 1892

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 962  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 776  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-v 1809

Copyright terms: Public domain