Theorem reximdvai 3015

Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 14-Nov-2002.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 8-Jan-2020.)

Hypothesis

Ref	Expression
reximdvai.1	|- ( ph -> ( x e. A -> ( ps -> ch ) ) )

Assertion

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

Distinct variable group:   ph, x

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

Proof of Theorem reximdvai

Step	Hyp	Ref	Expression
1		reximdvai.1	. . 3 |- ( ph -> ( x e. A -> ( ps -> ch ) ) )
2	1	ralrimiv 2965	. 2 |- ( ph -> A. x e. A ( ps -> ch ) )
3		rexim 3008	. 2 |- ( A. x e. A ( ps -> ch ) -> ( E. x e. A ps -> E. x e. A ch ) )
4	2, 3	syl 17	1 |- ( ph -> ( E. x e. A ps -> E. x e. A ch ) )

Syntax hints:   -> wi 4   e. wcel 1990   A.wral 2912   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:  reximdv  3016  reximdva  3017  reuind  3411  wefrc  5108  isomin  6587  isofrlem  6590  onfununi  7438  oaordex  7638  odi  7659  omass  7660  omeulem1  7662  noinfep  8557  rankwflemb  8656  infxpenlem  8836  coflim  9083  coftr  9095  zorn2lem7  9324  suplem1pr  9874  axpre-sup  9990  climbdd  14402  filufint  21724  cvati  29225  atcvat4i  29256  mdsymlem2  29263  mdsymlem3  29264  sumdmdii  29274  iccllysconn  31232  dftrpred3g  31733  incsequz2  33545  lcvat  34317  hlrelat3  34698  cvrval3  34699  cvrval4N  34700  2atlt  34725  cvrat4  34729  atbtwnexOLDN  34733  atbtwnex  34734  athgt  34742  2llnmat  34810  lnjatN  35066  2lnat  35070  cdlemb  35080  lhpexle3lem  35297  cdlemf1  35849  cdlemf2  35850  cdlemf  35851  cdlemk26b-3  36193  dvh4dimlem  36732  upbdrech  39519  limcperiod  39860  cncfshift  40087  cncfperiod  40092