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Theorem r19.45av 2541
 Description: Restricted version of one direction of Theorem 19.45 of [Margaris] p. 90. (The other direction doesn't hold when 𝐴 is empty.) (Contributed by NM, 2-Apr-2004.)
Assertion
Ref Expression
r19.45av (∃𝑥𝐴 (𝜑𝜓) → (𝜑 ∨ ∃𝑥𝐴 𝜓))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem r19.45av
StepHypRef Expression
1 r19.43 2539 . 2 (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓))
2 idd 21 . . . 4 (𝑥𝐴 → (𝜑𝜑))
32rexlimiv 2496 . . 3 (∃𝑥𝐴 𝜑𝜑)
43orim1i 715 . 2 ((∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓) → (𝜑 ∨ ∃𝑥𝐴 𝜓))
51, 4sylbi 120 1 (∃𝑥𝐴 (𝜑𝜓) → (𝜑 ∨ ∃𝑥𝐴 𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∨ wo 667   ∈ wcel 1445  ∃wrex 2371 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-4 1452  ax-17 1471  ax-ial 1479  ax-i5r 1480 This theorem depends on definitions:  df-bi 116  df-nf 1402  df-ral 2375  df-rex 2376 This theorem is referenced by: (None)
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