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Theorem 19.43 1592
Description: Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 2-Feb-2015.)
Assertion
Ref Expression
19.43 (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))

Proof of Theorem 19.43
StepHypRef Expression
1 hbe1 1456 . . . 4 (∃𝑥𝜑 → ∀𝑥𝑥𝜑)
2 hbe1 1456 . . . 4 (∃𝑥𝜓 → ∀𝑥𝑥𝜓)
31, 2hbor 1510 . . 3 ((∃𝑥𝜑 ∨ ∃𝑥𝜓) → ∀𝑥(∃𝑥𝜑 ∨ ∃𝑥𝜓))
4 19.8a 1554 . . . 4 (𝜑 → ∃𝑥𝜑)
5 19.8a 1554 . . . 4 (𝜓 → ∃𝑥𝜓)
64, 5orim12i 733 . . 3 ((𝜑𝜓) → (∃𝑥𝜑 ∨ ∃𝑥𝜓))
73, 6exlimih 1557 . 2 (∃𝑥(𝜑𝜓) → (∃𝑥𝜑 ∨ ∃𝑥𝜓))
8 orc 686 . . . 4 (𝜑 → (𝜑𝜓))
98eximi 1564 . . 3 (∃𝑥𝜑 → ∃𝑥(𝜑𝜓))
10 olc 685 . . . 4 (𝜓 → (𝜑𝜓))
1110eximi 1564 . . 3 (∃𝑥𝜓 → ∃𝑥(𝜑𝜓))
129, 11jaoi 690 . 2 ((∃𝑥𝜑 ∨ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))
137, 12impbii 125 1 (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  wb 104  wo 682  wex 1453
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 683  ax-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-ial 1499
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  19.44  1645  19.45  1646  19.34  1647  sborv  1846  r19.43  2566  rexun  3226  unipr  3720  uniun  3725  unopab  3977  dmun  4716  coundi  5010  coundir  5011
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