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Theorem 19.42h 1577
 Description: Theorem 19.42 of [Margaris] p. 90. New proofs should use 19.42 1578 instead. (Contributed by NM, 18-Aug-1993.) (New usage is discouraged.)
Hypothesis
Ref Expression
19.42h.1 (𝜑 → ∀𝑥𝜑)
Assertion
Ref Expression
19.42h (∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))

Proof of Theorem 19.42h
StepHypRef Expression
1 19.42h.1 . . 3 (𝜑 → ∀𝑥𝜑)
2119.41h 1575 . 2 (∃𝑥(𝜓𝜑) ↔ (∃𝑥𝜓𝜑))
3 exancom 1499 . 2 (∃𝑥(𝜑𝜓) ↔ ∃𝑥(𝜓𝜑))
4 ancom 253 . 2 ((𝜑 ∧ ∃𝑥𝜓) ↔ (∃𝑥𝜓𝜑))
52, 3, 43bitr4i 201 1 (∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1241  ∃wex 1381 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427 This theorem depends on definitions:  df-bi 110 This theorem is referenced by:  19.42v  1786
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