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Theorem exbi 1584
 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 biimp 117 . . . 4 ((𝜑𝜓) → (𝜑𝜓))
21alimi 1435 . . 3 (∀𝑥(𝜑𝜓) → ∀𝑥(𝜑𝜓))
3 exim 1579 . . 3 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
42, 3syl 14 . 2 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
5 biimpr 129 . . . 4 ((𝜑𝜓) → (𝜓𝜑))
65alimi 1435 . . 3 (∀𝑥(𝜑𝜓) → ∀𝑥(𝜓𝜑))
7 exim 1579 . . 3 (∀𝑥(𝜓𝜑) → (∃𝑥𝜓 → ∃𝑥𝜑))
86, 7syl 14 . 2 (∀𝑥(𝜑𝜓) → (∃𝑥𝜓 → ∃𝑥𝜑))
94, 8impbid 128 1 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1333  ∃wex 1472 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-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-ial 1514 This theorem depends on definitions:  df-bi 116 This theorem is referenced by:  exbii  1585  exbidh  1594  exintrbi  1613  19.19  1646  rexrnmpt  5607
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