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Theorem 19.41 1621
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 1567 . . 3 (∃𝑥(𝜑𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))
2 19.41.1 . . . . 5 𝑥𝜓
3219.9 1580 . . . 4 (∃𝑥𝜓𝜓)
43anbi2i 445 . . 3 ((∃𝑥𝜑 ∧ ∃𝑥𝜓) ↔ (∃𝑥𝜑𝜓))
51, 4sylib 120 . 2 (∃𝑥(𝜑𝜓) → (∃𝑥𝜑𝜓))
6 pm3.21 260 . . . 4 (𝜓 → (𝜑 → (𝜑𝜓)))
72, 6eximd 1548 . . 3 (𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))
87impcom 123 . 2 ((∃𝑥𝜑𝜓) → ∃𝑥(𝜑𝜓))
95, 8impbii 124 1 (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103  wnf 1394  wex 1426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-nf 1395
This theorem is referenced by:  19.42  1623  eean  1854  r19.41  2522  eliunxp  4575  dfopab2  5959  dfoprab3s  5960  xpcomco  6540
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