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Theorem r19.43 2589
Description: Restricted version of Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.) (Proof rewritten by Jim Kingdon, 5-Jun-2018.)
Assertion
Ref Expression
r19.43 (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓))

Proof of Theorem r19.43
StepHypRef Expression
1 df-rex 2422 . . . 4 (∃𝑥𝐴 (𝜑𝜓) ↔ ∃𝑥(𝑥𝐴 ∧ (𝜑𝜓)))
2 andi 807 . . . . 5 ((𝑥𝐴 ∧ (𝜑𝜓)) ↔ ((𝑥𝐴𝜑) ∨ (𝑥𝐴𝜓)))
32exbii 1584 . . . 4 (∃𝑥(𝑥𝐴 ∧ (𝜑𝜓)) ↔ ∃𝑥((𝑥𝐴𝜑) ∨ (𝑥𝐴𝜓)))
41, 3bitri 183 . . 3 (∃𝑥𝐴 (𝜑𝜓) ↔ ∃𝑥((𝑥𝐴𝜑) ∨ (𝑥𝐴𝜓)))
5 19.43 1607 . . 3 (∃𝑥((𝑥𝐴𝜑) ∨ (𝑥𝐴𝜓)) ↔ (∃𝑥(𝑥𝐴𝜑) ∨ ∃𝑥(𝑥𝐴𝜓)))
64, 5bitri 183 . 2 (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥(𝑥𝐴𝜑) ∨ ∃𝑥(𝑥𝐴𝜓)))
7 df-rex 2422 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
8 df-rex 2422 . . 3 (∃𝑥𝐴 𝜓 ↔ ∃𝑥(𝑥𝐴𝜓))
97, 8orbi12i 753 . 2 ((∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓) ↔ (∃𝑥(𝑥𝐴𝜑) ∨ ∃𝑥(𝑥𝐴𝜓)))
106, 9bitr4i 186 1 (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wo 697  wex 1468  wcel 1480  wrex 2417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514
This theorem depends on definitions:  df-bi 116  df-rex 2422
This theorem is referenced by:  r19.44av  2590  r19.45av  2591  r19.45mv  3456  r19.44mv  3457  iunun  3891  ltexprlemloc  7427
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