Theorem r19.29an 2511

Description: A commonly used pattern based on r19.29 2507. (Contributed by Thierry Arnoux, 29-Dec-2019.)

Hypothesis

Ref	Expression
r19.29an.1	|- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )

Assertion

Ref	Expression
r19.29an	|- ( ( ph /\ E. x e. A ps ) -> ch )

Distinct variable groups:   ch, x   ph, x

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

Proof of Theorem r19.29an

Step	Hyp	Ref	Expression
1		nfv 1467	. . 3 |- F/ x ph
2		nfre1 2420	. . 3 |- F/ x E. x e. A ps
3	1, 2	nfan 1503	. 2 |- F/ x ( ph /\ E. x e. A ps )
4		r19.29an.1	. . 3 |- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )
5	4	adantllr 466	. 2 |- ( ( ( ( ph /\ E. x e. A ps ) /\ x e. A ) /\ ps ) -> ch )
6		simpr 109	. 2 |- ( ( ph /\ E. x e. A ps ) -> E. x e. A ps )
7	3, 5, 6	r19.29af 2510	1 |- ( ( ph /\ E. x e. A ps ) -> ch )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 1439   E.wrex 2361

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-4 1446  ax-17 1465  ax-ial 1473  ax-i5r 1474

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-ral 2365  df-rex 2366

This theorem is referenced by:  isummolem2  10826