Theorem r19.29af2 2671

Description: A commonly used pattern based on r19.29 2668. (Contributed by Thierry Arnoux, 17-Dec-2017.)

Hypotheses

Ref	Expression
r19.29af2.p	|- F/ x ph
r19.29af2.c	|- F/ x ch
r19.29af2.1	|- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )
r19.29af2.2	|- ( ph -> E. x e. A ps )

Assertion

Ref	Expression
r19.29af2	|- ( ph -> ch )

Proof of Theorem r19.29af2

Step	Hyp	Ref	Expression
1		r19.29af2.2	. . 3 |- ( ph -> E. x e. A ps )
2		r19.29af2.p	. . . 4 |- F/ x ph
3		r19.29af2.1	. . . . 5 |- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )
4	3	exp31 364	. . . 4 |- ( ph -> ( x e. A -> ( ps -> ch ) ) )
5	2, 4	ralrimi 2601	. . 3 |- ( ph -> A. x e. A ( ps -> ch ) )
6	1, 5	jca 306	. 2 |- ( ph -> ( E. x e. A ps /\ A. x e. A ( ps -> ch ) ) )
7		r19.29r 2669	. 2 |- ( ( E. x e. A ps /\ A. x e. A ( ps -> ch ) ) -> E. x e. A ( ps /\ ( ps -> ch ) ) )
8		r19.29af2.c	. . 3 |- F/ x ch
9		pm3.35 347	. . . 4 |- ( ( ps /\ ( ps -> ch ) ) -> ch )
10	9	a1i 9	. . 3 |- ( x e. A -> ( ( ps /\ ( ps -> ch ) ) -> ch ) )
11	8, 10	rexlimi 2641	. 2 |- ( E. x e. A ( ps /\ ( ps -> ch ) ) -> ch )
12	6, 7, 11	3syl 17	1 |- ( ph -> ch )

Syntax hints:   -> wi 4   /\ wa 104   F/wnf 1506   e. wcel 2200   A.wral 2508   E.wrex 2509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-ral 2513  df-rex 2514

This theorem is referenced by:  r19.29af  2672  ctiunctlemfo  13010