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Theorem r19.44v 3343
 Description: One direction of a restricted quantifier version of 19.44 2241. The other direction holds when 𝐴 is nonempty, see r19.44zv 4432. (Contributed by NM, 2-Apr-2004.)
Assertion
Ref Expression
r19.44v (∃𝑥𝐴 (𝜑𝜓) → (∃𝑥𝐴 𝜑𝜓))
Distinct variable group:   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem r19.44v
StepHypRef Expression
1 r19.43 3342 . 2 (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓))
2 id 22 . . . 4 (𝜓𝜓)
32rexlimivw 3274 . . 3 (∃𝑥𝐴 𝜓𝜓)
43orim2i 908 . 2 ((∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓) → (∃𝑥𝐴 𝜑𝜓))
51, 4sylbi 220 1 (∃𝑥𝐴 (𝜑𝜓) → (∃𝑥𝐴 𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 844  ∃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 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-ral 3138  df-rex 3139 This theorem is referenced by: (None)
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