Theorem reximi 2627

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

Hypothesis

Ref	Expression
reximi.1	⊢ (𝜑 → 𝜓)

Assertion

Ref	Expression
reximi	⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)

Proof of Theorem reximi

Step	Hyp	Ref	Expression
1		reximi.1	. . 3 ⊢ (𝜑 → 𝜓)
2	1	a1i 9	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))
3	2	reximia 2625	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ∈ wcel 2200  ∃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  2993  ssiun  4007  iinss  4017  elunirn  5899  tfrcllemssrecs  6509  nnawordex  6688  iinerm  6767  erovlem  6787  xpf1o  7018  fidcenumlemim  7135  omniwomnimkv  7350  genprndl  7724  genprndu  7725  appdiv0nq  7767  ltexprlemm  7803  recexsrlem  7977  rereceu  8092  recexre  8741  aprcl  8809  rexanre  11752  climi2  11820  climi0  11821  climcaucn  11883  prodmodclem2  12109  prodmodc  12110  gcdsupex  12499  gcdsupcl  12500  bezoutlemeu  12549  dfgcd3  12552  isnsgrp  13460  rhmdvdsr  14160  eltg2b  14749  lmcvg  14912  cnptoprest  14934  lmtopcnp  14945  txbas  14953  metrest  15201  elply2  15430  2sqlem7  15821  umgr2edg1  16028  umgr2edgneu  16031  bj-charfunbi  16283  bj-findis  16451