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Theorem 19.43 1881
Description: Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.)
Assertion
Ref Expression
19.43 (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))

Proof of Theorem 19.43
StepHypRef Expression
1 df-or 848 . . . 4 ((𝜑𝜓) ↔ (¬ 𝜑𝜓))
21exbii 1847 . . 3 (∃𝑥(𝜑𝜓) ↔ ∃𝑥𝜑𝜓))
3 19.35 1876 . . 3 (∃𝑥𝜑𝜓) ↔ (∀𝑥 ¬ 𝜑 → ∃𝑥𝜓))
4 alnex 1780 . . . 4 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
54imbi1i 349 . . 3 ((∀𝑥 ¬ 𝜑 → ∃𝑥𝜓) ↔ (¬ ∃𝑥𝜑 → ∃𝑥𝜓))
62, 3, 53bitri 297 . 2 (∃𝑥(𝜑𝜓) ↔ (¬ ∃𝑥𝜑 → ∃𝑥𝜓))
7 df-or 848 . 2 ((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ (¬ ∃𝑥𝜑 → ∃𝑥𝜓))
86, 7bitr4i 278 1 (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wo 847  wal 1537  wex 1778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808
This theorem depends on definitions:  df-bi 207  df-or 848  df-ex 1779
This theorem is referenced by:  19.34  1985  19.44v  1991  19.45v  1992  19.44  2236  19.45  2237  eeor  2334  rexun  4195  uniprg  4922  uniun  4929  unopab  5223  zfpair  5420  dmun  5920  dmopab2rex  5927  coundi  6266  coundir  6267  kmlem16  10207  vdwapun  17013  satfdm  35375  satf0op  35383  dmopab3rexdif  35411  bj-nnfor  36752  bj-nnford  36753  exor  42682  pm10.42  44388
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