Theorem bj-vtoclf 32908

Description: Remove dependency on ax-ext 2602, df-clab 2609 and df-cleq 2615 (and df-sb 1881 and df-v 3202) from vtoclf 3258. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-vtoclf.nf	|- F/ x ps
bj-vtoclf.s	|- A e. V
bj-vtoclf.maj	|- ( x = A -> ( ph <-> ps ) )
bj-vtoclf.min	|- ph

Assertion

Ref	Expression
bj-vtoclf	|- ps

Distinct variable groups:   x, A   x, V

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

Proof of Theorem bj-vtoclf

Step	Hyp	Ref	Expression
1		bj-vtoclf.nf	. . 3 |- F/ x ps
2		bj-vtoclf.s	. . . . 5 |- A e. V
3	2	bj-issetiv 32863	. . . 4 |- E. x x = A
4		bj-vtoclf.maj	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
5	4	biimpd 219	. . . 4 |- ( x = A -> ( ph -> ps ) )
6	3, 5	eximii 1764	. . 3 |- E. x ( ph -> ps )
7	1, 6	19.36i 2099	. 2 |- ( A. x ph -> ps )
8		bj-vtoclf.min	. 2 |- ph
9	7, 8	mpg 1724	1 |- ps

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   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-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-nf 1710  df-clel 2618

This theorem is referenced by:  bj-vtocl  32909