Theorem bj-vtoclg1fv 32912

Description: Version of bj-vtoclg1f 32911 with a dv condition on x , V. This removes dependency on df-sb 1881 and df-clab 2609. Prefer its use over bj-vtoclg1f 32911 when sufficient (in particular when V is substituted for _V). (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-vtoclg1fv.nf	|- F/ x ps
bj-vtoclg1fv.maj	|- ( x = A -> ( ph -> ps ) )
bj-vtoclg1fv.min	|- ph

Assertion

Ref	Expression
bj-vtoclg1fv	|- ( A e. V -> ps )

Distinct variable groups:   x, A   x, V

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

Proof of Theorem bj-vtoclg1fv

Step	Hyp	Ref	Expression
1		bj-elissetv 32861	. 2 |- ( A e. V -> E. x x = A )
2		bj-vtoclg1fv.nf	. . 3 |- F/ x ps
3		bj-vtoclg1fv.maj	. . 3 |- ( x = A -> ( ph -> ps ) )
4		bj-vtoclg1fv.min	. . 3 |- ph
5	2, 3, 4	bj-exlimmpi 32905	. 2 |- ( E. x x = A -> ps )
6	1, 5	syl 17	1 |- ( A e. V -> ps )

Syntax hints:   -> wi 4   = 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-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-clel 2618

This theorem is referenced by: (None)