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Theorem 19.42 2143
Description: Theorem 19.42 of [Margaris] p. 90. See 19.42v 1921 for a version requiring fewer axioms. See exan 1828 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 2141 . 2 (∃𝑥(𝜓𝜑) ↔ (∃𝑥𝜓𝜑))
3 exancom 1827 . 2 (∃𝑥(𝜑𝜓) ↔ ∃𝑥(𝜓𝜑))
4 ancom 465 . 2 ((𝜑 ∧ ∃𝑥𝜓) ↔ (∃𝑥𝜓𝜑))
52, 3, 43bitr4i 292 1 (∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wex 1744  wnf 1748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-nf 1750
This theorem is referenced by:  eean  2217  bnj916  31129  bnj983  31147
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