Theorem reximi 2627

Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 18-Oct-1996.)

Hypothesis

Ref	Expression
reximi.1	|- ( ph -> ps )

Assertion

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

Proof of Theorem reximi

Step	Hyp	Ref	Expression
1		reximi.1	. . 3 |- ( ph -> ps )
2	1	a1i 9	. 2 |- ( x e. A -> ( ph -> ps ) )
3	2	reximia 2625	1 |- ( E. x e. A ph -> E. x e. A ps )

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-ial 1580

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

This theorem is referenced by:  rexanaliim  2636  r19.29d2r  2675  r19.35-1  2681  r19.40  2685  reu3  2994  ssiun  4010  iinss  4020  elunirn  5902  tfrcllemssrecs  6513  nnawordex  6692  iinerm  6771  erovlem  6791  xpf1o  7025  fidcenumlemim  7142  omniwomnimkv  7357  genprndl  7731  genprndu  7732  appdiv0nq  7774  ltexprlemm  7810  recexsrlem  7984  rereceu  8099  recexre  8748  aprcl  8816  rexanre  11771  climi2  11839  climi0  11840  climcaucn  11902  prodmodclem2  12128  prodmodc  12129  gcdsupex  12518  gcdsupcl  12519  bezoutlemeu  12568  dfgcd3  12571  isnsgrp  13479  rhmdvdsr  14179  eltg2b  14768  lmcvg  14931  cnptoprest  14953  lmtopcnp  14964  txbas  14972  metrest  15220  elply2  15449  2sqlem7  15840  umgr2edg1  16048  umgr2edgneu  16051  bj-charfunbi  16342  bj-findis  16510