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Theorem r19.43 3120
Description: Restricted quantifier version of 19.43 1880. (Contributed by NM, 27-May-1998.) (Proof shortened by Andrew Salmon, 30-May-2011.)
Assertion
Ref Expression
r19.43 (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓))

Proof of Theorem r19.43
StepHypRef Expression
1 r19.35 3106 . 2 (∃𝑥𝐴𝜑𝜓) ↔ (∀𝑥𝐴 ¬ 𝜑 → ∃𝑥𝐴 𝜓))
2 df-or 848 . . 3 ((𝜑𝜓) ↔ (¬ 𝜑𝜓))
32rexbii 3092 . 2 (∃𝑥𝐴 (𝜑𝜓) ↔ ∃𝑥𝐴𝜑𝜓))
4 df-or 848 . . 3 ((∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓) ↔ (¬ ∃𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
5 ralnex 3070 . . . 4 (∀𝑥𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴 𝜑)
65imbi1i 349 . . 3 ((∀𝑥𝐴 ¬ 𝜑 → ∃𝑥𝐴 𝜓) ↔ (¬ ∃𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
74, 6bitr4i 278 . 2 ((∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓) ↔ (∀𝑥𝐴 ¬ 𝜑 → ∃𝑥𝐴 𝜓))
81, 3, 73bitr4i 303 1 (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wo 847  wral 3059  wrex 3068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-ral 3060  df-rex 3069
This theorem is referenced by:  r19.45v  3191  r19.44v  3192  r19.45zv  4509  r19.44zv  4510  iunun  5098  soseq  8183  wemapsolem  9588  pythagtriplem2  16851  pythagtrip  16868  dcubic  26904  addsdilem1  28192  mulsasslem2  28205  legtrid  28614  axcontlem4  28997  erdszelem11  35186  satfvsucsuc  35350  fmla1  35372  seglelin  36098  hashnexinjle  42111  fimgmcyclem  42520  rexor  42655  diophun  42761  rexzrexnn0  42792  dfvopnbgr2  47777  dfsclnbgr6  47782  ldepslinc  48355
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