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  7451  ctssdc  7454  suplocexprlemru  8087  suplocexprlemloc  8089  suplocsrlemb  8174  aptap  8981  4sqlemffi  13198  4sqleminfi  13199  4sqexercise2  13201  4sqlemsdc  13202  ennnfonelemhom  13358  gzsumfzval  13764  innei  15355  ivthinclemlr  15829  ivthinclemur  15831  limccnpcntop  15867  limccoap  15870  2lgslem1c  16375  2lgslem3a1  16382  2lgslem3b1  16383  2lgslem3c1  16384  2lgslem3d1  16385  umgrnloop  16523  stnot  17205