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Theorem r19.35-1 2457
Description: Restricted quantifier version of 19.35-1 1515. (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 2447 . . 3 ((∀𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 (𝜑𝜓)) → ∃𝑥𝐴 (𝜑 ∧ (𝜑𝜓)))
2 pm3.35 329 . . . 4 ((𝜑 ∧ (𝜑𝜓)) → 𝜓)
32reximi 2413 . . 3 (∃𝑥𝐴 (𝜑 ∧ (𝜑𝜓)) → ∃𝑥𝐴 𝜓)
41, 3syl 14 . 2 ((∀𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 (𝜑𝜓)) → ∃𝑥𝐴 𝜓)
54expcom 109 1 (∃𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wral 2303  wrex 2304
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-ial 1427
This theorem depends on definitions:  df-bi 110  df-ral 2308  df-rex 2309
This theorem is referenced by:  r19.36av  2458  r19.37  2459  iinexgm  3904  bndndx  8114
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