Theorem rexlimdvaa 2669

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 375	. 2 |- ( ( ph /\ x e. A ) -> ( ps -> ch ) )
3	2	rexlimdva 2668	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:  rexlimddv  2673  nnsucuniel  6768  omp1eomlem  7434  ctmlemr  7448  mulgt0sr  8145  axpre-suploclemres  8268  cnegex  8505  receuap  9001  recapb  9003  rexanuz  11768  climcaucn  12133  fsumiun  12260  dvdsval2  12573  nninfctlemfo  12833  prmind2  12914  nn0sqdcq  13004  sqrtrirr  13005  pcprmpw2  13132  pockthg  13156  dvdsrvald  14449  dvdsrd  14450  dvdsrex  14454  unitgrp  14472  isnzr2  14540  znunit  15043  tgcl  15214  neiint  15295  restopnb  15331  iscnp4  15368  blssexps  15579  blssex  15580  lgsne0  16255  lgsquadlem1  16294