Theorem bj-vtoclgfALT 33021

Description: Alternate proof of vtoclgf 3264. Proof from vtoclgft 3254. (This may have been the original proof before shortening.) (Contributed by BJ, 30-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bj-vtoclgfALT.1	|- F/_ x A
bj-vtoclgfALT.2	|- F/ x ps
bj-vtoclgfALT.3	|- ( x = A -> ( ph <-> ps ) )
bj-vtoclgfALT.4	|- ph

Assertion

Ref	Expression
bj-vtoclgfALT	|- ( A e. V -> ps )

Proof of Theorem bj-vtoclgfALT

Step	Hyp	Ref	Expression
1		bj-vtoclgfALT.1	. . 3 |- F/_ x A
2		bj-vtoclgfALT.2	. . 3 |- F/ x ps
3	1, 2	pm3.2i 471	. 2 |- ( F/_ x A /\ F/ x ps )
4		bj-vtoclgfALT.3	. . . 4 |- ( x = A -> ( ph <-> ps ) )
5	4	ax-gen 1722	. . 3 |- A. x ( x = A -> ( ph <-> ps ) )
6		bj-vtoclgfALT.4	. . . 4 |- ph
7	6	ax-gen 1722	. . 3 |- A. x ph
8	5, 7	pm3.2i 471	. 2 |- ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph )
9		vtoclgft 3254	. 2 |- ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) /\ A e. V ) -> ps )
10	3, 8, 9	mp3an12 1414	1 |- ( A e. V -> ps )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   F/wnf 1708   e. wcel 1990   F/_wnfc 2751

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-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  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-nfc 2753  df-v 3202

This theorem is referenced by: (None)