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Theorem r19.37v 3279
 Description: Restricted quantifier version of one direction of 19.37v 1948. (The other direction holds iff 𝐴 is nonempty, see r19.37zv 4324.) (Contributed by NM, 2-Apr-2004.) Reduce axiom usage. (Revised by Wolf Lammen, 18-Jun-2023.)
Assertion
Ref Expression
r19.37v (∃𝑥𝐴 (𝜑𝜓) → (𝜑 → ∃𝑥𝐴 𝜓))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem r19.37v
StepHypRef Expression
1 id 22 . . 3 (𝜑𝜑)
21ralrimivw 3127 . 2 (𝜑 → ∀𝑥𝐴 𝜑)
3 r19.35 3276 . . 3 (∃𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
43biimpi 208 . 2 (∃𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
52, 4syl5 34 1 (∃𝑥𝐴 (𝜑𝜓) → (𝜑 → ∃𝑥𝐴 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wral 3082  ∃wrex 3083 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869 This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1743  df-ral 3087  df-rex 3088 This theorem is referenced by:  ssiun  4830  isucn2  22585
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