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Theorem 19.40 1595
Description: Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
19.40 (∃𝑥(𝜑𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))

Proof of Theorem 19.40
StepHypRef Expression
1 exsimpl 1581 . 2 (∃𝑥(𝜑𝜓) → ∃𝑥𝜑)
2 simpr 109 . . 3 ((𝜑𝜓) → 𝜓)
32eximi 1564 . 2 (∃𝑥(𝜑𝜓) → ∃𝑥𝜓)
41, 3jca 304 1 (∃𝑥(𝜑𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wex 1453
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 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-ial 1499
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  19.40-2  1596  19.41h  1648  19.41  1649  exdistrfor  1756  uniin  3726  copsexg  4136  dmin  4717  imadif  5173  imainlem  5174
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