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Theorem r19.36vf 40068
Description: Restricted quantifier version of one direction of 19.36 2265. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypothesis
Ref Expression
r19.36vf.1 𝑥𝜓
Assertion
Ref Expression
r19.36vf (∃𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑𝜓))

Proof of Theorem r19.36vf
StepHypRef Expression
1 r19.35 3263 . 2 (∃𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
2 r19.36vf.1 . . . 4 𝑥𝜓
3 idd 24 . . . 4 (𝑥𝐴 → (𝜓𝜓))
42, 3rexlimi 3203 . . 3 (∃𝑥𝐴 𝜓𝜓)
54imim2i 16 . 2 ((∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓) → (∀𝑥𝐴 𝜑𝜓))
61, 5sylbi 209 1 (∃𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wnf 1879  wcel 2157  wral 3087  wrex 3088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-12 2213
This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876  df-nf 1880  df-ral 3092  df-rex 3093
This theorem is referenced by:  iinssf  40071
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