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Theorem 19.42 2237
 Description: Theorem 19.42 of [Margaris] p. 90. See 19.42v 1954 for a version requiring fewer axioms. See exan 1863 for an immediate version. (Contributed by NM, 18-Aug-1993.)
Hypothesis
Ref Expression
19.42.1 𝑥𝜑
Assertion
Ref Expression
19.42 (∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))

Proof of Theorem 19.42
StepHypRef Expression
1 19.42.1 . . 3 𝑥𝜑
2119.41 2236 . 2 (∃𝑥(𝜓𝜑) ↔ (∃𝑥𝜓𝜑))
3 exancom 1862 . 2 (∃𝑥(𝜑𝜓) ↔ ∃𝑥(𝜓𝜑))
4 ancom 464 . 2 ((𝜑 ∧ ∃𝑥𝜓) ↔ (∃𝑥𝜓𝜑))
52, 3, 43bitr4i 306 1 (∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399  ∃wex 1781  Ⅎwnf 1785 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2176 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786 This theorem is referenced by:  eean  2361  bnj596  32131  bnj916  32319  bnj983  32337
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