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Theorem 19.42 2202
Description: Theorem 19.42 of [Margaris] p. 90. See 19.42v 1932 for a version requiring fewer axioms. See exan 1844 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 2201 . 2 (∃𝑥(𝜓𝜑) ↔ (∃𝑥𝜓𝜑))
3 exancom 1843 . 2 (∃𝑥(𝜑𝜓) ↔ ∃𝑥(𝜓𝜑))
4 ancom 461 . 2 ((𝜑 ∧ ∃𝑥𝜓) ↔ (∃𝑥𝜓𝜑))
52, 3, 43bitr4i 304 1 (∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wex 1762  wnf 1766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-12 2140
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1763  df-nf 1767
This theorem is referenced by:  eean  2324  bnj596  31626  bnj916  31813  bnj983  31831
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