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Theorem 19.43 1588
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 1452 . . . 4 (∃𝑥𝜑 → ∀𝑥𝑥𝜑)
2 hbe1 1452 . . . 4 (∃𝑥𝜓 → ∀𝑥𝑥𝜓)
31, 2hbor 1506 . . 3 ((∃𝑥𝜑 ∨ ∃𝑥𝜓) → ∀𝑥(∃𝑥𝜑 ∨ ∃𝑥𝜓))
4 19.8a 1550 . . . 4 (𝜑 → ∃𝑥𝜑)
5 19.8a 1550 . . . 4 (𝜓 → ∃𝑥𝜓)
64, 5orim12i 731 . . 3 ((𝜑𝜓) → (∃𝑥𝜑 ∨ ∃𝑥𝜓))
73, 6exlimih 1553 . 2 (∃𝑥(𝜑𝜓) → (∃𝑥𝜑 ∨ ∃𝑥𝜓))
8 orc 684 . . . 4 (𝜑 → (𝜑𝜓))
98eximi 1560 . . 3 (∃𝑥𝜑 → ∃𝑥(𝜑𝜓))
10 olc 683 . . . 4 (𝜓 → (𝜑𝜓))
1110eximi 1560 . . 3 (∃𝑥𝜓 → ∃𝑥(𝜑𝜓))
129, 11jaoi 688 . 2 ((∃𝑥𝜑 ∨ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))
137, 12impbii 125 1 (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  wb 104  wo 680  wex 1449
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-4 1468  ax-ial 1495
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  19.44  1641  19.45  1642  19.34  1643  sborv  1842  r19.43  2561  rexun  3220  unipr  3714  uniun  3719  unopab  3965  dmun  4704  coundi  4996  coundir  4997
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