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Theorem pm5.71 973
 Description: Theorem *5.71 of [WhiteheadRussell] p. 125. (Contributed by Roy F. Longton, 23-Jun-2005.)
Assertion
Ref Expression
pm5.71 ((𝜓 → ¬ 𝜒) → (((𝜑𝜓) ∧ 𝜒) ↔ (𝜑𝜒)))

Proof of Theorem pm5.71
StepHypRef Expression
1 orel2 397 . . . 4 𝜓 → ((𝜑𝜓) → 𝜑))
2 orc 399 . . . 4 (𝜑 → (𝜑𝜓))
31, 2impbid1 214 . . 3 𝜓 → ((𝜑𝜓) ↔ 𝜑))
43anbi1d 737 . 2 𝜓 → (((𝜑𝜓) ∧ 𝜒) ↔ (𝜑𝜒)))
5 pm2.21 119 . . 3 𝜒 → (𝜒 → ((𝜑𝜓) ↔ 𝜑)))
65pm5.32rd 670 . 2 𝜒 → (((𝜑𝜓) ∧ 𝜒) ↔ (𝜑𝜒)))
74, 6ja 172 1 ((𝜓 → ¬ 𝜒) → (((𝜑𝜓) ∧ 𝜒) ↔ (𝜑𝜒)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ 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)
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