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Theorem r19.45zv 4420
 Description: Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
Assertion
Ref Expression
r19.45zv (𝐴 ≠ ∅ → (∃𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∨ ∃𝑥𝐴 𝜓)))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem r19.45zv
StepHypRef Expression
1 r19.9rzv 4417 . . 3 (𝐴 ≠ ∅ → (𝜑 ↔ ∃𝑥𝐴 𝜑))
21orbi1d 914 . 2 (𝐴 ≠ ∅ → ((𝜑 ∨ ∃𝑥𝐴 𝜓) ↔ (∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓)))
3 r19.43 3332 . 2 (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓))
42, 3syl6rbbr 293 1 (𝐴 ≠ ∅ → (∃𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∨ ∃𝑥𝐴 𝜓)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∨ wo 844   ≠ wne 3011  ∃wrex 3131  ∅c0 4265 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  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-11 2161  ax-12 2178  ax-ext 2794 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-dif 3911  df-nul 4266 This theorem is referenced by: (None)
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