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Theorem reximdv2 2529
 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 1852 . 2 (𝜑 → (∃𝑥(𝑥𝐴𝜓) → ∃𝑥(𝑥𝐵𝜒)))
3 df-rex 2420 . 2 (∃𝑥𝐴 𝜓 ↔ ∃𝑥(𝑥𝐴𝜓))
4 df-rex 2420 . 2 (∃𝑥𝐵 𝜒 ↔ ∃𝑥(𝑥𝐵𝜒))
52, 3, 43imtr4g 204 1 (𝜑 → (∃𝑥𝐴 𝜓 → ∃𝑥𝐵 𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103  ∃wex 1468   ∈ wcel 1480  ∃wrex 2415 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514 This theorem depends on definitions:  df-bi 116  df-rex 2420 This theorem is referenced by:  reximssdv  2534  ssrexv  3157  ssimaex  5475  ico0  10032  ioc0  10033  r19.2uz  10758  trilpolemlt1  13223
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