MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  19.42-1 Structured version   Visualization version   GIF version

Theorem 19.42-1 2142
Description: One direction of 19.42 2143. (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 pm3.2 462 . . 3 (𝜑 → (𝜓 → (𝜑𝜓)))
31, 2eximd 2123 . 2 (𝜑 → (∃𝑥𝜓 → ∃𝑥(𝜑𝜓)))
43imp 444 1 ((𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wex 1744  wnf 1748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-nf 1750
This theorem is referenced by:  bnj596  30942
  Copyright terms: Public domain W3C validator