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Theorem 19.41 1590
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 1536 . . 3 (∃𝑥(𝜑𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))
2 19.41.1 . . . . 5 𝑥𝜓
3219.9 1549 . . . 4 (∃𝑥𝜓𝜓)
43anbi2i 438 . . 3 ((∃𝑥𝜑 ∧ ∃𝑥𝜓) ↔ (∃𝑥𝜑𝜓))
51, 4sylib 131 . 2 (∃𝑥(𝜑𝜓) → (∃𝑥𝜑𝜓))
6 pm3.21 255 . . . 4 (𝜓 → (𝜑 → (𝜑𝜓)))
72, 6eximd 1517 . . 3 (𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))
87impcom 120 . 2 ((∃𝑥𝜑𝜓) → ∃𝑥(𝜑𝜓))
95, 8impbii 121 1 (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102  wnf 1363  wex 1395
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 1350  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-4 1414  ax-ial 1441
This theorem depends on definitions:  df-bi 114  df-nf 1364
This theorem is referenced by:  19.42  1592  eean  1820  r19.41  2480  eliunxp  4500  dfopab2  5840  dfoprab3s  5841  xpcomco  6328
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