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

Theorem 19.42-1OLD 2271
 Description: One direction of 19.42 2272. Obsolete as of 9-Oct-2022. (Contributed by Wolf Lammen, 10-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
19.42.1 𝑥𝜑
Assertion
Ref Expression
19.42-1OLD ((𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))

Proof of Theorem 19.42-1OLD
StepHypRef Expression
1 19.42.1 . . 3 𝑥𝜑
2 pm3.2 462 . . 3 (𝜑 → (𝜓 → (𝜑𝜓)))
31, 2eximd 2251 . 2 (𝜑 → (∃𝑥𝜓 → ∃𝑥(𝜑𝜓)))
43imp 396 1 ((𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 385  ∃wex 1875  Ⅎwnf 1879 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-12 2213 This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876  df-nf 1880 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator