Theorem bj-exlimmp 13650

Description: Lemma for bj-vtoclgf 13657. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-exlimmp.nf	|- F/ x ps
bj-exlimmp.min	|- ( ch -> ph )

Assertion

Ref	Expression
bj-exlimmp	|- ( A. x ( ch -> ( ph -> ps ) ) -> ( E. x ch -> ps ) )

Proof of Theorem bj-exlimmp

Step	Hyp	Ref	Expression
1		nfa1 1529	. 2 |- F/ x A. x ( ch -> ( ph -> ps ) )
2		bj-exlimmp.nf	. 2 |- F/ x ps
3		bj-exlimmp.min	. . . . 5 |- ( ch -> ph )
4		idd 21	. . . . 5 |- ( ch -> ( ps -> ps ) )
5	3, 4	embantd 56	. . . 4 |- ( ch -> ( ( ph -> ps ) -> ps ) )
6	5	a2i 11	. . 3 |- ( ( ch -> ( ph -> ps ) ) -> ( ch -> ps ) )
7	6	sps 1525	. 2 |- ( A. x ( ch -> ( ph -> ps ) ) -> ( ch -> ps ) )
8	1, 2, 7	exlimd 1585	1 |- ( A. x ( ch -> ( ph -> ps ) ) -> ( E. x ch -> ps ) )

Syntax hints:   -> wi 4   A.wal 1341   F/wnf 1448   E.wex 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie2 1482  ax-4 1498  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-nf 1449

This theorem is referenced by:  bj-vtoclgft  13656