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Theorem reximdv2 2418
 Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 17-Sep-2003.)
Hypothesis
Ref Expression
reximdv2.1 (𝜑 → ((𝑥𝐴𝜓) → (𝑥𝐵𝜒)))
Assertion
Ref Expression
reximdv2 (𝜑 → (∃𝑥𝐴 𝜓 → ∃𝑥𝐵 𝜒))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem reximdv2
StepHypRef Expression
1 reximdv2.1 . . 3 (𝜑 → ((𝑥𝐴𝜓) → (𝑥𝐵𝜒)))
21eximdv 1760 . 2 (𝜑 → (∃𝑥(𝑥𝐴𝜓) → ∃𝑥(𝑥𝐵𝜒)))
3 df-rex 2312 . 2 (∃𝑥𝐴 𝜓 ↔ ∃𝑥(𝑥𝐴𝜓))
4 df-rex 2312 . 2 (∃𝑥𝐵 𝜒 ↔ ∃𝑥(𝑥𝐵𝜒))
52, 3, 43imtr4g 194 1 (𝜑 → (∃𝑥𝐴 𝜓 → ∃𝑥𝐵 𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∃wex 1381   ∈ 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-rex 2312 This theorem is referenced by:  ssrexv  3005  ssimaex  5234  r19.2uz  9591
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