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Theorem reximdd 41282
Description: Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
Hypotheses
Ref Expression
reximdd.1 𝑥𝜑
reximdd.2 ((𝜑𝑥𝐴𝜓) → 𝜒)
reximdd.3 (𝜑 → ∃𝑥𝐴 𝜓)
Assertion
Ref Expression
reximdd (𝜑 → ∃𝑥𝐴 𝜒)

Proof of Theorem reximdd
StepHypRef Expression
1 reximdd.3 . 2 (𝜑 → ∃𝑥𝐴 𝜓)
2 reximdd.1 . . 3 𝑥𝜑
3 reximdd.2 . . . 4 ((𝜑𝑥𝐴𝜓) → 𝜒)
433exp 1113 . . 3 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
52, 4reximdai 3315 . 2 (𝜑 → (∃𝑥𝐴 𝜓 → ∃𝑥𝐴 𝜒))
61, 5mpd 15 1 (𝜑 → ∃𝑥𝐴 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1081  wnf 1777  wcel 2107  wrex 3143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-12 2169
This theorem depends on definitions:  df-bi 208  df-an 397  df-3an 1083  df-ex 1774  df-nf 1778  df-ral 3147  df-rex 3148
This theorem is referenced by:  xlimmnfvlem2  41975  xlimmnfv  41976  xlimpnfvlem2  41979  xlimpnfv  41980
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