Theorem rexlimdva2 39339

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 630	. 2 |- ( ph -> ( x e. A -> ( ps -> ch ) ) )
3	2	rexlimdv 3030	1 |- ( ph -> ( E. x e. A ps -> ch ) )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   E.wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-ral 2917  df-rex 2918

This theorem is referenced by:  supminfxr  39694  infrpgernmpt  39695  limsupresxr  39998  liminfresxr  39999  liminflelimsuplem  40007  limsupgtlem  40009  liminfvalxr  40015  liminfreuzlem  40034  cnrefiisplem  40055  xlimmnfvlem2  40059  xlimpnfvlem2  40063  smfliminflem  41036