Theorem reximdv 2631

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 2630	1 |- ( ph -> ( E. x e. A ps -> E. x e. A ch ) )

Syntax hints:   -> wi 4   e. wcel 2200   E.wrex 2509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-ral 2513  df-rex 2514

This theorem is referenced by:  r19.12  2637  reusv3  4551  rexxfrd  4554  iunpw  4571  fvelima  5685  carden2bex  7362  prnmaddl  7677  prarloclem5  7687  prarloc2  7691  genprndl  7708  genprndu  7709  ltpopr  7782  recexprlemm  7811  recexprlemopl  7812  recexprlemopu  7814  recexprlem1ssl  7820  recexprlem1ssu  7821  cauappcvgprlemupu  7836  caucvgprlemupu  7859  caucvgprprlemupu  7887  caucvgsrlemoffres  7987  map2psrprg  7992  resqrexlemgt0  11531  subcn2  11822  bezoutlembz  12525  pythagtriplem19  12805  mplsubgfileminv  14664  tgcl  14738  neiss  14824  ssnei2  14831  tgcnp  14883  cnptopco  14896  cnptopresti  14912  lmtopcnp  14924  blssexps  15103  blssex  15104  mopni3  15158  neibl  15165  metss  15168  metcnp3  15185  mpomulcn  15240  rescncf  15255  limcresi  15340  plyss  15412  umgrnloop0  15917  uhgr2edg  16004