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Theorem 19.41 2233
Description: Theorem 19.41 of [Margaris] p. 90. See 19.41v 1947 for a version requiring fewer axioms. (Contributed by NM, 14-May-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 1884 . . 3 (∃𝑥(𝜑𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))
2 19.41.1 . . . . 5 𝑥𝜓
3219.9 2203 . . . 4 (∃𝑥𝜓𝜓)
43anbi2i 623 . . 3 ((∃𝑥𝜑 ∧ ∃𝑥𝜓) ↔ (∃𝑥𝜑𝜓))
51, 4sylib 218 . 2 (∃𝑥(𝜑𝜓) → (∃𝑥𝜑𝜓))
6 pm3.21 471 . . . 4 (𝜓 → (𝜑 → (𝜑𝜓)))
72, 6eximd 2214 . . 3 (𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))
87impcom 407 . 2 ((∃𝑥𝜑𝜓) → ∃𝑥(𝜑𝜓))
95, 8impbii 209 1 (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wex 1776  wnf 1780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-nf 1781
This theorem is referenced by:  19.42  2234  eean  2349  eeeanv  2351  equsexALT  2422  2sb5rf  2475  r19.41  3261  eliunxp  5851  dfopab2  8076  dfoprab3s  8077  xpcomco  9101  mpomptxf  32694  bnj605  34900  bnj607  34909  2sb5nd  44558  2sb5ndVD  44908  2sb5ndALT  44930  eliunxp2  48179
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