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Theorem r19.44zv 4262
Description: Restricted version of Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
Assertion
Ref Expression
r19.44zv (𝐴 ≠ ∅ → (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓)))
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem r19.44zv
StepHypRef Expression
1 r19.9rzv 4258 . . 3 (𝐴 ≠ ∅ → (𝜓 ↔ ∃𝑥𝐴 𝜓))
21orbi2d 940 . 2 (𝐴 ≠ ∅ → ((∃𝑥𝐴 𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓)))
3 r19.43 3274 . 2 (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓))
42, 3syl6rbbr 282 1 (𝐴 ≠ ∅ → (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wo 874  wne 2971  wrex 3090  c0 4115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-v 3387  df-dif 3772  df-nul 4116
This theorem is referenced by: (None)
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