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Theorem 19.42 1594
Description: Theorem 19.42 of [Margaris] p. 90. (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 1592 . 2 (∃𝑥(𝜓𝜑) ↔ (∃𝑥𝜓𝜑))
3 exancom 1515 . 2 (∃𝑥(𝜑𝜓) ↔ ∃𝑥(𝜓𝜑))
4 ancom 257 . 2 ((𝜑 ∧ ∃𝑥𝜓) ↔ (∃𝑥𝜓𝜑))
52, 3, 43bitr4i 205 1 (∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102  wnf 1365  wex 1397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-nf 1366
This theorem is referenced by:  eean  1822  r2exf  2359
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