Theorem rexlimdva2 2671

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 2667	1 |- ( ph -> ( E. x e. A ps -> ch ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   e. wcel 2209   E.wrex 2529

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-i5r 1588

This proof depends on definitions:  df-bi 117  df-nf 1514  df-ral 2533  df-rex 2534

This theorem is used by:  ctssdclemn0  7450  ctssdc  7453  suplocexprlemru  8086  suplocexprlemloc  8088  suplocsrlemb  8173  aptap  8978  4sqlemffi  13175  4sqleminfi  13176  4sqexercise2  13178  4sqlemsdc  13179  ennnfonelemhom  13306  gzsumfzval  13711  innei  15264  ivthinclemlr  15738  ivthinclemur  15740  limccnpcntop  15776  limccoap  15779  2lgslem1c  16209  2lgslem3a1  16216  2lgslem3b1  16217  2lgslem3c1  16218  2lgslem3d1  16219  umgrnloop  16357  stnot  17039