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Theorem 19.42-1 2107
Description: One direction of 19.42 2108. (Contributed by Wolf Lammen, 10-Jul-2021.)
Hypothesis
Ref Expression
19.42.1 𝑥𝜑
Assertion
Ref Expression
19.42-1 ((𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))

Proof of Theorem 19.42-1
StepHypRef Expression
1 19.42.1 . . 3 𝑥𝜑
2 pm3.2 463 . . 3 (𝜑 → (𝜓 → (𝜑𝜓)))
31, 2eximd 2088 . 2 (𝜑 → (∃𝑥𝜓 → ∃𝑥(𝜑𝜓)))
43imp 445 1 ((𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wex 1701  wnf 1705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-12 2049
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-nf 1707
This theorem is referenced by:  bnj596  30516
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