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Theorem r19.44av 2588
Description: One direction of a restricted quantifier version of Theorem 19.44 of [Margaris] p. 90. The other direction doesn't hold when 𝐴 is empty. (Contributed by NM, 2-Apr-2004.)
Assertion
Ref Expression
r19.44av (∃𝑥𝐴 (𝜑𝜓) → (∃𝑥𝐴 𝜑𝜓))
Distinct variable group:   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem r19.44av
StepHypRef Expression
1 r19.43 2587 . 2 (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓))
2 idd 21 . . . 4 (𝑥𝐴 → (𝜓𝜓))
32rexlimiv 2541 . . 3 (∃𝑥𝐴 𝜓𝜓)
43orim2i 750 . 2 ((∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓) → (∃𝑥𝐴 𝜑𝜓))
51, 4sylbi 120 1 (∃𝑥𝐴 (𝜑𝜓) → (∃𝑥𝐴 𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 697  wcel 1480  wrex 2415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-ral 2419  df-rex 2420
This theorem is referenced by: (None)
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