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Theorem reximd2a 3007
Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by Thierry Arnoux, 27-Jan-2020.)
Hypotheses
Ref Expression
reximd2a.1 𝑥𝜑
reximd2a.2 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝑥𝐵)
reximd2a.3 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
reximd2a.4 (𝜑 → ∃𝑥𝐴 𝜓)
Assertion
Ref Expression
reximd2a (𝜑 → ∃𝑥𝐵 𝜒)

Proof of Theorem reximd2a
StepHypRef Expression
1 reximd2a.4 . 2 (𝜑 → ∃𝑥𝐴 𝜓)
2 reximd2a.1 . . . 4 𝑥𝜑
3 reximd2a.2 . . . . . 6 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝑥𝐵)
4 reximd2a.3 . . . . . 6 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
53, 4jca 554 . . . . 5 (((𝜑𝑥𝐴) ∧ 𝜓) → (𝑥𝐵𝜒))
65expl 647 . . . 4 (𝜑 → ((𝑥𝐴𝜓) → (𝑥𝐵𝜒)))
72, 6eximd 2083 . . 3 (𝜑 → (∃𝑥(𝑥𝐴𝜓) → ∃𝑥(𝑥𝐵𝜒)))
8 df-rex 2913 . . 3 (∃𝑥𝐴 𝜓 ↔ ∃𝑥(𝑥𝐴𝜓))
9 df-rex 2913 . . 3 (∃𝑥𝐵 𝜒 ↔ ∃𝑥(𝑥𝐵𝜒))
107, 8, 93imtr4g 285 . 2 (𝜑 → (∃𝑥𝐴 𝜓 → ∃𝑥𝐵 𝜒))
111, 10mpd 15 1 (𝜑 → ∃𝑥𝐵 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wex 1701  wnf 1705  wcel 1987  wrex 2908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-nf 1707  df-rex 2913
This theorem is referenced by:  locfinreflem  29689
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