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Theorem 19.43 1535
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 1400 . . . 4 (∃𝑥𝜑 → ∀𝑥𝑥𝜑)
2 hbe1 1400 . . . 4 (∃𝑥𝜓 → ∀𝑥𝑥𝜓)
31, 2hbor 1454 . . 3 ((∃𝑥𝜑 ∨ ∃𝑥𝜓) → ∀𝑥(∃𝑥𝜑 ∨ ∃𝑥𝜓))
4 19.8a 1498 . . . 4 (𝜑 → ∃𝑥𝜑)
5 19.8a 1498 . . . 4 (𝜓 → ∃𝑥𝜓)
64, 5orim12i 686 . . 3 ((𝜑𝜓) → (∃𝑥𝜑 ∨ ∃𝑥𝜓))
73, 6exlimih 1500 . 2 (∃𝑥(𝜑𝜓) → (∃𝑥𝜑 ∨ ∃𝑥𝜓))
8 orc 643 . . . 4 (𝜑 → (𝜑𝜓))
98eximi 1507 . . 3 (∃𝑥𝜑 → ∃𝑥(𝜑𝜓))
10 olc 642 . . . 4 (𝜓 → (𝜑𝜓))
1110eximi 1507 . . 3 (∃𝑥𝜓 → ∃𝑥(𝜑𝜓))
129, 11jaoi 646 . 2 ((∃𝑥𝜑 ∨ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))
137, 12impbii 121 1 (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  wb 102  wo 639  wex 1397
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
This theorem is referenced by:  19.44  1588  19.45  1589  19.34  1590  sborv  1786  r19.43  2485  rexun  3151  unipr  3622  uniun  3627  unopab  3864  dmun  4570  coundi  4850  coundir  4851
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