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Theorem r19.35-1 2581
Description: Restricted quantifier version of 19.35-1 1603. (Contributed by Jim Kingdon, 4-Jun-2018.)
Assertion
Ref Expression
r19.35-1 (∃𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))

Proof of Theorem r19.35-1
StepHypRef Expression
1 r19.29 2569 . . 3 ((∀𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 (𝜑𝜓)) → ∃𝑥𝐴 (𝜑 ∧ (𝜑𝜓)))
2 pm3.35 344 . . . 4 ((𝜑 ∧ (𝜑𝜓)) → 𝜓)
32reximi 2529 . . 3 (∃𝑥𝐴 (𝜑 ∧ (𝜑𝜓)) → ∃𝑥𝐴 𝜓)
41, 3syl 14 . 2 ((∀𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 (𝜑𝜓)) → ∃𝑥𝐴 𝜓)
54expcom 115 1 (∃𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wral 2416  wrex 2417
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-ial 1514
This theorem depends on definitions:  df-bi 116  df-ral 2421  df-rex 2422
This theorem is referenced by:  r19.36av  2582  r19.37  2583  iinexgm  4079  bndndx  8983
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