Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ex-natded9.20-2 Structured version   Visualization version   GIF version

Theorem ex-natded9.20-2 26667
 Description: A more efficient proof of Theorem 9.20 of [Clemente] p. 45. Compare with ex-natded9.20 26666. (Contributed by David A. Wheeler, 19-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
ex-natded9.20.1 (𝜑 → (𝜓 ∧ (𝜒𝜃)))
Assertion
Ref Expression
ex-natded9.20-2 (𝜑 → ((𝜓𝜒) ∨ (𝜓𝜃)))

Proof of Theorem ex-natded9.20-2
StepHypRef Expression
1 ex-natded9.20.1 . . . . 5 (𝜑 → (𝜓 ∧ (𝜒𝜃)))
21simpld 474 . . . 4 (𝜑𝜓)
32anim1i 590 . . 3 ((𝜑𝜒) → (𝜓𝜒))
43orcd 406 . 2 ((𝜑𝜒) → ((𝜓𝜒) ∨ (𝜓𝜃)))
52anim1i 590 . . 3 ((𝜑𝜃) → (𝜓𝜃))
65olcd 407 . 2 ((𝜑𝜃) → ((𝜓𝜒) ∨ (𝜓𝜃)))
71simprd 478 . 2 (𝜑 → (𝜒𝜃))
84, 6, 7mpjaodan 823 1 (𝜑 → ((𝜓𝜒) ∨ (𝜓𝜃)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator