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Theorem reximia 2462
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 2461 . 2 (∀𝑥𝐴 (𝜑𝜓) → (∃𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
2 reximia.1 . 2 (𝑥𝐴 → (𝜑𝜓))
31, 2mprg 2426 1 (∃𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1434  wrex 2354
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-ral 2358  df-rex 2359
This theorem is referenced by:  reximi  2464  iunpw  4265  nsmallnqq  6874  1idprl  7052  1idpru  7053  qmulz  9003  zq  9006  caubnd2  10377
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