Theorem r19.29an 2574

Description: A commonly used pattern based on r19.29 2569. (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 1508	. . 3 |- F/ x ph
2		nfre1 2476	. . 3 |- F/ x E. x e. A ps
3	1, 2	nfan 1544	. 2 |- F/ x ( ph /\ E. x e. A ps )
4		r19.29an.1	. . 3 |- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )
5	4	adantllr 472	. 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 2573	1 |- ( ( ph /\ E. x e. A ps ) -> ch )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 1480   E.wrex 2417

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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-ral 2421  df-rex 2422

This theorem is referenced by:  summodclem2  11154