Theorem reximdv 2571

Description: Deduction from Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version with strong hypothesis.) (Contributed by NM, 24-Jun-1998.)

Hypothesis

Ref	Expression
reximdv.1	|- ( ph -> ( ps -> ch ) )

Assertion

Ref	Expression
reximdv	|- ( 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 reximdv

Step	Hyp	Ref	Expression
1		reximdv.1	. . 3 |- ( ph -> ( ps -> ch ) )
2	1	a1d 22	. 2 |- ( ph -> ( x e. A -> ( ps -> ch ) ) )
3	2	reximdvai 2570	1 |- ( ph -> ( E. x e. A ps -> E. x e. A ch ) )

Syntax hints:   -> wi 4   e. wcel 2141   E.wrex 2449

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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-ral 2453  df-rex 2454

This theorem is referenced by:  r19.12  2576  reusv3  4443  rexxfrd  4446  iunpw  4463  fvelima  5546  carden2bex  7153  prnmaddl  7439  prarloclem5  7449  prarloc2  7453  genprndl  7470  genprndu  7471  ltpopr  7544  recexprlemm  7573  recexprlemopl  7574  recexprlemopu  7576  recexprlem1ssl  7582  recexprlem1ssu  7583  cauappcvgprlemupu  7598  caucvgprlemupu  7621  caucvgprprlemupu  7649  caucvgsrlemoffres  7749  map2psrprg  7754  resqrexlemgt0  10971  subcn2  11261  bezoutlembz  11946  pythagtriplem19  12223  tgcl  12817  neiss  12903  ssnei2  12910  tgcnp  12962  cnptopco  12975  cnptopresti  12991  lmtopcnp  13003  blssexps  13182  blssex  13183  mopni3  13237  neibl  13244  metss  13247  metcnp3  13264  rescncf  13321  limcresi  13388