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Theorem pm5.75 947
 Description: Theorem *5.75 of [WhiteheadRussell] p. 126. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 23-Dec-2012.)
Assertion
Ref Expression
pm5.75 (((𝜒 → ¬ 𝜓) ∧ (𝜑 ↔ (𝜓𝜒))) → ((𝜑 ∧ ¬ 𝜓) ↔ 𝜒))

Proof of Theorem pm5.75
StepHypRef Expression
1 anbi1 462 . . 3 ((𝜑 ↔ (𝜓𝜒)) → ((𝜑 ∧ ¬ 𝜓) ↔ ((𝜓𝜒) ∧ ¬ 𝜓)))
2 orcom 718 . . . . 5 ((𝜓𝜒) ↔ (𝜒𝜓))
32anbi1i 454 . . . 4 (((𝜓𝜒) ∧ ¬ 𝜓) ↔ ((𝜒𝜓) ∧ ¬ 𝜓))
4 pm5.61 784 . . . 4 (((𝜒𝜓) ∧ ¬ 𝜓) ↔ (𝜒 ∧ ¬ 𝜓))
53, 4bitri 183 . . 3 (((𝜓𝜒) ∧ ¬ 𝜓) ↔ (𝜒 ∧ ¬ 𝜓))
61, 5syl6bb 195 . 2 ((𝜑 ↔ (𝜓𝜒)) → ((𝜑 ∧ ¬ 𝜓) ↔ (𝜒 ∧ ¬ 𝜓)))
7 pm4.71 387 . . . 4 ((𝜒 → ¬ 𝜓) ↔ (𝜒 ↔ (𝜒 ∧ ¬ 𝜓)))
87biimpi 119 . . 3 ((𝜒 → ¬ 𝜓) → (𝜒 ↔ (𝜒 ∧ ¬ 𝜓)))
98bicomd 140 . 2 ((𝜒 → ¬ 𝜓) → ((𝜒 ∧ ¬ 𝜓) ↔ 𝜒))
106, 9sylan9bbr 459 1 (((𝜒 → ¬ 𝜓) ∧ (𝜑 ↔ (𝜓𝜒))) → ((𝜑 ∧ ¬ 𝜓) ↔ 𝜒))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 605  ax-io 699 This theorem depends on definitions:  df-bi 116 This theorem is referenced by: (None)
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