Theorem bj-exlimmp 9909

Description: Lemma for bj-vtoclgf 9915. (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 1434	. 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 50	. . . 4 |- ( ch -> ( ( ph -> ps ) -> ps ) )
6	5	a2i 11	. . 3 |- ( ( ch -> ( ph -> ps ) ) -> ( ch -> ps ) )
7	6	sps 1430	. 2 |- ( A. x ( ch -> ( ph -> ps ) ) -> ( ch -> ps ) )
8	1, 2, 7	exlimd 1488	1 |- ( A. x ( ch -> ( ph -> ps ) ) -> ( E. x ch -> ps ) )

Syntax hints:   -> wi 4   A.wal 1241   F/wnf 1349   E.wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie2 1383  ax-4 1400  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-nf 1350

This theorem is referenced by:  bj-vtoclgft  9914