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Theorem reximia 2431
Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 10-Feb-1997.)
Hypothesis
Ref Expression
reximia.1 (𝑥𝐴 → (𝜑𝜓))
Assertion
Ref Expression
reximia (∃𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓)

Proof of Theorem reximia
StepHypRef Expression
1 rexim 2430 . 2 (∀𝑥𝐴 (𝜑𝜓) → (∃𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
2 reximia.1 . 2 (𝑥𝐴 → (𝜑𝜓))
31, 2mprg 2395 1 (∃𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1409  wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-ral 2328  df-rex 2329
This theorem is referenced by:  reximi  2433  iunpw  4238  nsmallnqq  6567  1idprl  6745  1idpru  6746  qmulz  8654  zq  8657  caubnd2  9943
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