Theorem rexlimdva2 2651

Description: Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypothesis

Ref	Expression
rexlimdva2.1	|- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )

Assertion

Ref	Expression
rexlimdva2	|- ( ph -> ( E. x e. A ps -> ch ) )

Distinct variable groups:   ch, x   ph, x

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

Proof of Theorem rexlimdva2

Step	Hyp	Ref	Expression
1		rexlimdva2.1	. . 3 |- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )
2	1	exp31 364	. 2 |- ( ph -> ( x e. A -> ( ps -> ch ) ) )
3	2	rexlimdv 2647	1 |- ( ph -> ( E. x e. A ps -> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2200   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-nf 1507  df-ral 2513  df-rex 2514

This theorem is referenced by:  ctssdclemn0  7273  ctssdc  7276  suplocexprlemru  7902  suplocexprlemloc  7904  suplocsrlemb  7989  aptap  8793  4sqlemffi  12914  4sqleminfi  12915  4sqexercise2  12917  4sqlemsdc  12918  ennnfonelemhom  12981  gsumfzval  13419  reldvdsrsrg  14050  innei  14831  ivthinclemlr  15305  ivthinclemur  15307  limccnpcntop  15343  limccoap  15346  2lgslem1c  15763  2lgslem3a1  15770  2lgslem3b1  15771  2lgslem3c1  15772  2lgslem3d1  15773  umgrnloop  15910