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Theorem exbi 1568
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 1416 . . 3 (∀𝑥(𝜑𝜓) → ∀𝑥(𝜑𝜓))
3 exim 1563 . . 3 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
42, 3syl 14 . 2 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
5 bi2 129 . . . 4 ((𝜑𝜓) → (𝜓𝜑))
65alimi 1416 . . 3 (∀𝑥(𝜑𝜓) → ∀𝑥(𝜓𝜑))
7 exim 1563 . . 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 1314  wex 1453
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 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-ial 1499
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  exbii  1569  exbidh  1578  exintrbi  1597  19.19  1629  rexrnmpt  5531
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