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Theorem exbi 1540
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 116 . . . 4 ((𝜑𝜓) → (𝜑𝜓))
21alimi 1389 . . 3 (∀𝑥(𝜑𝜓) → ∀𝑥(𝜑𝜓))
3 exim 1535 . . 3 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
42, 3syl 14 . 2 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
5 bi2 128 . . . 4 ((𝜑𝜓) → (𝜓𝜑))
65alimi 1389 . . 3 (∀𝑥(𝜑𝜓) → ∀𝑥(𝜓𝜑))
7 exim 1535 . . 3 (∀𝑥(𝜓𝜑) → (∃𝑥𝜓 → ∃𝑥𝜑))
86, 7syl 14 . 2 (∀𝑥(𝜑𝜓) → (∃𝑥𝜓 → ∃𝑥𝜑))
94, 8impbid 127 1 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1287  wex 1426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-ial 1472
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  exbii  1541  exbidh  1550  exintrbi  1569  19.19  1601  rexrnmpt  5442
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