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Theorem reximia 2414
 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 2413 . 2 (∀𝑥𝐴 (𝜑𝜓) → (∃𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
2 reximia.1 . 2 (𝑥𝐴 → (𝜑𝜓))
31, 2mprg 2378 1 (∃𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1393  ∃wrex 2307 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427 This theorem depends on definitions:  df-bi 110  df-ral 2311  df-rex 2312 This theorem is referenced by:  reximi  2416  iunpw  4211  nsmallnqq  6510  1idprl  6688  1idpru  6689  qmulz  8558  zq  8561  caubnd2  9713
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