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Theorem r19.37 3325
 Description: Restricted quantifier version of one direction of 19.37 2235. (The other direction does not hold when 𝐴 is empty.) (Contributed by FL, 13-May-2012.) (Revised by Mario Carneiro, 11-Dec-2016.)
Hypothesis
Ref Expression
r19.37.1 𝑥𝜑
Assertion
Ref Expression
r19.37 (∃𝑥𝐴 (𝜑𝜓) → (𝜑 → ∃𝑥𝐴 𝜓))

Proof of Theorem r19.37
StepHypRef Expression
1 r19.35 3323 . 2 (∃𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
2 r19.37.1 . . . 4 𝑥𝜑
3 ax-1 6 . . . 4 (𝜑 → (𝑥𝐴𝜑))
42, 3ralrimi 3205 . . 3 (𝜑 → ∀𝑥𝐴 𝜑)
54imim1i 63 . 2 ((∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓) → (𝜑 → ∃𝑥𝐴 𝜓))
61, 5sylbi 220 1 (∃𝑥𝐴 (𝜑𝜓) → (𝜑 → ∃𝑥𝐴 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  Ⅎwnf 1785   ∈ wcel 2114  ∀wral 3130  ∃wrex 3131 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 1911  ax-6 1970  ax-7 2015  ax-12 2178 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-ral 3135  df-rex 3136 This theorem is referenced by:  ss2iundf  40290
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