Theorem rexlimdvaa 2575

Description: Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Mario Carneiro, 15-Jun-2016.)

Hypothesis

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

Assertion

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

Distinct variable groups:   ph, x   ch, x

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

Proof of Theorem rexlimdvaa

Step	Hyp	Ref	Expression
1		rexlimdvaa.1	. . 3 |- ( ( ph /\ ( x e. A /\ ps ) ) -> ch )
2	1	expr 373	. 2 |- ( ( ph /\ x e. A ) -> ( ps -> ch ) )
3	2	rexlimdva 2574	1 |- ( ph -> ( E. x e. A ps -> ch ) )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2128   E.wrex 2436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-17 1506  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-ral 2440  df-rex 2441

This theorem is referenced by:  rexlimddv  2579  nnsucuniel  6442  omp1eomlem  7038  ctmlemr  7052  mulgt0sr  7698  axpre-suploclemres  7821  cnegex  8053  receuap  8543  rexanuz  10888  climcaucn  11248  fsumiun  11374  dvdsval2  11686  prmind2  11997  tgcl  12475  neiint  12556  restopnb  12592  iscnp4  12629  blssexps  12840  blssex  12841