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Theorem exbi 1495
 Description: Theorem 19.18 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
exbi (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓))

Proof of Theorem exbi
StepHypRef Expression
1 bi1 111 . . . 4 ((𝜑𝜓) → (𝜑𝜓))
21alimi 1344 . . 3 (∀𝑥(𝜑𝜓) → ∀𝑥(𝜑𝜓))
3 exim 1490 . . 3 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
42, 3syl 14 . 2 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
5 bi2 121 . . . 4 ((𝜑𝜓) → (𝜓𝜑))
65alimi 1344 . . 3 (∀𝑥(𝜑𝜓) → ∀𝑥(𝜓𝜑))
7 exim 1490 . . 3 (∀𝑥(𝜓𝜑) → (∃𝑥𝜓 → ∃𝑥𝜑))
86, 7syl 14 . 2 (∀𝑥(𝜑𝜓) → (∃𝑥𝜓 → ∃𝑥𝜑))
94, 8impbid 120 1 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241  ∃wex 1381 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427 This theorem depends on definitions:  df-bi 110 This theorem is referenced by:  exbii  1496  exbidh  1505  exintrbi  1524  19.19  1556  rexrnmpt  5310
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