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Theorem reximdvai 2419
 Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 14-Nov-2002.)
Hypothesis
Ref Expression
reximdvai.1 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
Assertion
Ref Expression
reximdvai (𝜑 → (∃𝑥𝐴 𝜓 → ∃𝑥𝐴 𝜒))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)

Proof of Theorem reximdvai
StepHypRef Expression
1 nfv 1421 . 2 𝑥𝜑
2 reximdvai.1 . 2 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
31, 2reximdai 2417 1 (𝜑 → (∃𝑥𝐴 𝜓 → ∃𝑥𝐴 𝜒))
 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-17 1419  ax-ial 1427 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-ral 2311  df-rex 2312 This theorem is referenced by:  reximdv  2420  reximdva  2421  reuind  2744
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