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Theorem r19.45v 3334
Description: Restricted quantifier version of one direction of 19.45 2241. The other direction holds when 𝐴 is nonempty, see r19.45zv 4420. (Contributed by NM, 2-Apr-2004.)
Assertion
Ref Expression
r19.45v (∃𝑥𝐴 (𝜑𝜓) → (𝜑 ∨ ∃𝑥𝐴 𝜓))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem r19.45v
StepHypRef Expression
1 r19.43 3332 . 2 (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓))
2 id 22 . . . 4 (𝜑𝜑)
32rexlimivw 3268 . . 3 (∃𝑥𝐴 𝜑𝜑)
43orim1i 907 . 2 ((∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓) → (𝜑 ∨ ∃𝑥𝐴 𝜓))
51, 4sylbi 220 1 (∃𝑥𝐴 (𝜑𝜓) → (𝜑 ∨ ∃𝑥𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 844  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
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-ral 3135  df-rex 3136
This theorem is referenced by: (None)
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