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  7456  ctssdc  7459  suplocexprlemru  8092  suplocexprlemloc  8094  suplocsrlemb  8179  aptap  8986  4sqlemffi  13203  4sqleminfi  13204  4sqexercise2  13206  4sqlemsdc  13207  ennnfonelemhom  13363  gzsumfzval  13769  psrbagmaxlem1  15178  innei  15406  ivthinclemlr  15880  ivthinclemur  15882  limccnpcntop  15918  limccoap  15921  2lgslem1c  16431  2lgslem3a1  16438  2lgslem3b1  16439  2lgslem3c1  16440  2lgslem3d1  16441  umgrnloop  16579  stnot  17261