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Theorem 19.19 1599
Description: Theorem 19.19 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
Hypothesis
Ref Expression
19.19.1 𝑥𝜑
Assertion
Ref Expression
19.19 (∀𝑥(𝜑𝜓) → (𝜑 ↔ ∃𝑥𝜓))

Proof of Theorem 19.19
StepHypRef Expression
1 19.19.1 . . 3 𝑥𝜑
2119.9 1578 . 2 (∃𝑥𝜑𝜑)
3 exbi 1538 . 2 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓))
42, 3syl5bbr 192 1 (∀𝑥(𝜑𝜓) → (𝜑 ↔ ∃𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1285  wnf 1392  wex 1424
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 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-ial 1470
This theorem depends on definitions:  df-bi 115  df-nf 1393
This theorem is referenced by: (None)
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