Theorem reximdv 2634

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 2633	1 |- ( ph -> ( E. x e. A ps -> E. x e. A ch ) )

Syntax hints:   -> wi 4   e. wcel 2202   E.wrex 2512

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-ral 2516  df-rex 2517

This theorem is referenced by:  r19.12  2640  reusv3  4563  rexxfrd  4566  iunpw  4583  fvelima  5706  carden2bex  7454  prnmaddl  7770  prarloclem5  7780  prarloc2  7784  genprndl  7801  genprndu  7802  ltpopr  7875  recexprlemm  7904  recexprlemopl  7905  recexprlemopu  7907  recexprlem1ssl  7913  recexprlem1ssu  7914  cauappcvgprlemupu  7929  caucvgprlemupu  7952  caucvgprprlemupu  7980  caucvgsrlemoffres  8080  map2psrprg  8085  resqrexlemgt0  11660  subcn2  11951  bezoutlembz  12655  pythagtriplem19  12935  mplsubgfileminv  14801  tgcl  14875  neiss  14961  ssnei2  14968  tgcnp  15020  cnptopco  15033  cnptopresti  15049  lmtopcnp  15061  blssexps  15240  blssex  15241  mopni3  15295  neibl  15302  metss  15305  metcnp3  15322  mpomulcn  15377  rescncf  15392  limcresi  15477  plyss  15549  umgrnloop0  16058  uhgr2edg  16147