Theorem r19.29af2 2610

Description: A commonly used pattern based on r19.29 2607. (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 362	. . . 4 |- ( ph -> ( x e. A -> ( ps -> ch ) ) )
5	2, 4	ralrimi 2541	. . 3 |- ( ph -> A. x e. A ( ps -> ch ) )
6	1, 5	jca 304	. 2 |- ( ph -> ( E. x e. A ps /\ A. x e. A ( ps -> ch ) ) )
7		r19.29r 2608	. 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		pm3.35 345	. . . 4 |- ( ( ps /\ ( ps -> ch ) ) -> ch )
10	9	a1i 9	. . 3 |- ( x e. A -> ( ( ps /\ ( ps -> ch ) ) -> ch ) )
11	8, 10	rexlimi 2580	. 2 |- ( E. x e. A ( ps /\ ( ps -> ch ) ) -> ch )
12	6, 7, 11	3syl 17	1 |- ( ph -> ch )

Syntax hints:   -> wi 4   /\ wa 103   F/wnf 1453   e. wcel 2141   A.wral 2448   E.wrex 2449

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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-ral 2453  df-rex 2454

This theorem is referenced by:  r19.29af  2611  ctiunctlemfo  12394