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Theorem exbi 1566
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 117 . . . 4 ((𝜑𝜓) → (𝜑𝜓))
21alimi 1414 . . 3 (∀𝑥(𝜑𝜓) → ∀𝑥(𝜑𝜓))
3 exim 1561 . . 3 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
42, 3syl 14 . 2 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
5 bi2 129 . . . 4 ((𝜑𝜓) → (𝜓𝜑))
65alimi 1414 . . 3 (∀𝑥(𝜑𝜓) → ∀𝑥(𝜓𝜑))
7 exim 1561 . . 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 1312  wex 1451
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-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-ial 1497
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  exbii  1567  exbidh  1576  exintrbi  1595  19.19  1627  rexrnmpt  5517
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