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Theorem r19.36vf 41695
 Description: Restricted quantifier version of one direction of 19.36 2234. (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 3332 . 2 (∃𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
2 r19.36vf.1 . . . 4 𝑥𝜓
3 idd 24 . . . 4 (𝑥𝐴 → (𝜓𝜓))
42, 3rexlimi 3307 . . 3 (∃𝑥𝐴 𝜓𝜓)
54imim2i 16 . 2 ((∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓) → (∀𝑥𝐴 𝜑𝜓))
61, 5sylbi 220 1 (∃𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  Ⅎwnf 1785   ∈ wcel 2115  ∀wral 3133  ∃wrex 3134 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-12 2179 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-ral 3138  df-rex 3139 This theorem is referenced by:  iinssf  41698
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