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Theorem r19.43 3121
Description: Restricted quantifier version of 19.43 1881. (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 3107 . 2 (∃𝑥𝐴𝜑𝜓) ↔ (∀𝑥𝐴 ¬ 𝜑 → ∃𝑥𝐴 𝜓))
2 df-or 848 . . 3 ((𝜑𝜓) ↔ (¬ 𝜑𝜓))
32rexbii 3093 . 2 (∃𝑥𝐴 (𝜑𝜓) ↔ ∃𝑥𝐴𝜑𝜓))
4 df-or 848 . . 3 ((∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓) ↔ (¬ ∃𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
5 ralnex 3071 . . . 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 3060  wrex 3069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1779  df-ral 3061  df-rex 3070
This theorem is referenced by:  r19.45v  3192  r19.44v  3193  r19.45zv  4502  r19.44zv  4503  iunun  5092  soseq  8185  wemapsolem  9591  pythagtriplem2  16856  pythagtrip  16873  dcubic  26890  addsdilem1  28178  mulsasslem2  28191  legtrid  28600  axcontlem4  28983  erdszelem11  35207  satfvsucsuc  35371  fmla1  35393  seglelin  36118  hashnexinjle  42131  fimgmcyclem  42548  rexor  42683  diophun  42789  rexzrexnn0  42820  dfvopnbgr2  47844  dfsclnbgr6  47849  ldepslinc  48431
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