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Theorem r19.43 2485
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 2329 . . . 4 (∃𝑥𝐴 (𝜑𝜓) ↔ ∃𝑥(𝑥𝐴 ∧ (𝜑𝜓)))
2 andi 742 . . . . 5 ((𝑥𝐴 ∧ (𝜑𝜓)) ↔ ((𝑥𝐴𝜑) ∨ (𝑥𝐴𝜓)))
32exbii 1512 . . . 4 (∃𝑥(𝑥𝐴 ∧ (𝜑𝜓)) ↔ ∃𝑥((𝑥𝐴𝜑) ∨ (𝑥𝐴𝜓)))
41, 3bitri 177 . . 3 (∃𝑥𝐴 (𝜑𝜓) ↔ ∃𝑥((𝑥𝐴𝜑) ∨ (𝑥𝐴𝜓)))
5 19.43 1535 . . 3 (∃𝑥((𝑥𝐴𝜑) ∨ (𝑥𝐴𝜓)) ↔ (∃𝑥(𝑥𝐴𝜑) ∨ ∃𝑥(𝑥𝐴𝜓)))
64, 5bitri 177 . 2 (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥(𝑥𝐴𝜑) ∨ ∃𝑥(𝑥𝐴𝜓)))
7 df-rex 2329 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
8 df-rex 2329 . . 3 (∃𝑥𝐴 𝜓 ↔ ∃𝑥(𝑥𝐴𝜓))
97, 8orbi12i 691 . 2 ((∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓) ↔ (∃𝑥(𝑥𝐴𝜑) ∨ ∃𝑥(𝑥𝐴𝜓)))
106, 9bitr4i 180 1 (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓))
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102  wo 639  wex 1397  wcel 1409  wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-rex 2329
This theorem is referenced by:  r19.44av  2486  r19.45av  2487  r19.45mv  3342  iunun  3761  ltexprlemloc  6762
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