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Theorem 19.41 1647
Description: Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-Jan-2018.)
Hypothesis
Ref Expression
19.41.1 𝑥𝜓
Assertion
Ref Expression
19.41 (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Proof of Theorem 19.41
StepHypRef Expression
1 19.40 1593 . . 3 (∃𝑥(𝜑𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))
2 19.41.1 . . . . 5 𝑥𝜓
3219.9 1606 . . . 4 (∃𝑥𝜓𝜓)
43anbi2i 450 . . 3 ((∃𝑥𝜑 ∧ ∃𝑥𝜓) ↔ (∃𝑥𝜑𝜓))
51, 4sylib 121 . 2 (∃𝑥(𝜑𝜓) → (∃𝑥𝜑𝜓))
6 pm3.21 262 . . . 4 (𝜓 → (𝜑 → (𝜑𝜓)))
72, 6eximd 1574 . . 3 (𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))
87impcom 124 . 2 ((∃𝑥𝜑𝜓) → ∃𝑥(𝜑𝜓))
95, 8impbii 125 1 (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wnf 1419  wex 1451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-ial 1497
This theorem depends on definitions:  df-bi 116  df-nf 1420
This theorem is referenced by:  19.42  1649  eean  1881  r19.41  2561  eliunxp  4646  dfopab2  6053  dfoprab3s  6054  xpcomco  6686
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