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Theorem ex-natded9.20-2 26433
Description: A more efficient proof of Theorem 9.20 of [Clemente] p. 45. Compare with ex-natded9.20 26432. (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 473 . . . 4 (𝜑𝜓)
32anim1i 589 . . 3 ((𝜑𝜒) → (𝜓𝜒))
43orcd 405 . 2 ((𝜑𝜒) → ((𝜓𝜒) ∨ (𝜓𝜃)))
52anim1i 589 . . 3 ((𝜑𝜃) → (𝜓𝜃))
65olcd 406 . 2 ((𝜑𝜃) → ((𝜓𝜒) ∨ (𝜓𝜃)))
71simprd 477 . 2 (𝜑 → (𝜒𝜃))
84, 6, 7mpjaodan 822 1 (𝜑 → ((𝜓𝜒) ∨ (𝜓𝜃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 381  wa 382
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 195  df-or 383  df-an 384
This theorem is referenced by: (None)
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