Theorem vtoclegft 3280

Description: Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 3281.) (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)

Assertion

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

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   B( x)

Proof of Theorem vtoclegft

Step	Hyp	Ref	Expression
1		elisset 3215	. . . 4 |- ( A e. B -> E. x x = A )
2		exim 1761	. . . 4 |- ( A. x ( x = A -> ph ) -> ( E. x x = A -> E. x ph ) )
3	1, 2	mpan9 486	. . 3 |- ( ( A e. B /\ A. x ( x = A -> ph ) ) -> E. x ph )
4	3	3adant2 1080	. 2 |- ( ( A e. B /\ F/ x ph /\ A. x ( x = A -> ph ) ) -> E. x ph )
5		19.9t 2071	. . 3 |- ( F/ x ph -> ( E. x ph <-> ph ) )
6	5	3ad2ant2 1083	. 2 |- ( ( A e. B /\ F/ x ph /\ A. x ( x = A -> ph ) ) -> ( E. x ph <-> ph ) )
7	4, 6	mpbid 222	1 |- ( ( A e. B /\ F/ x ph /\ A. x ( x = A -> ph ) ) -> ph )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   A.wal 1481   = wceq 1483   E.wex 1704   F/wnf 1708   e. wcel 1990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202

This theorem is referenced by:  vtoclefex  33181