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Theorem pm5.75 906
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 454 . . 3 ((𝜑 ↔ (𝜓𝜒)) → ((𝜑 ∧ ¬ 𝜓) ↔ ((𝜓𝜒) ∧ ¬ 𝜓)))
2 orcom 680 . . . . 5 ((𝜓𝜒) ↔ (𝜒𝜓))
32anbi1i 446 . . . 4 (((𝜓𝜒) ∧ ¬ 𝜓) ↔ ((𝜒𝜓) ∧ ¬ 𝜓))
4 pm5.61 741 . . . 4 (((𝜒𝜓) ∧ ¬ 𝜓) ↔ (𝜒 ∧ ¬ 𝜓))
53, 4bitri 182 . . 3 (((𝜓𝜒) ∧ ¬ 𝜓) ↔ (𝜒 ∧ ¬ 𝜓))
61, 5syl6bb 194 . 2 ((𝜑 ↔ (𝜓𝜒)) → ((𝜑 ∧ ¬ 𝜓) ↔ (𝜒 ∧ ¬ 𝜓)))
7 pm4.71 381 . . . 4 ((𝜒 → ¬ 𝜓) ↔ (𝜒 ↔ (𝜒 ∧ ¬ 𝜓)))
87biimpi 118 . . 3 ((𝜒 → ¬ 𝜓) → (𝜒 ↔ (𝜒 ∧ ¬ 𝜓)))
98bicomd 139 . 2 ((𝜒 → ¬ 𝜓) → ((𝜒 ∧ ¬ 𝜓) ↔ 𝜒))
106, 9sylan9bbr 451 1 (((𝜒 → ¬ 𝜓) ∧ (𝜑 ↔ (𝜓𝜒))) → ((𝜑 ∧ ¬ 𝜓) ↔ 𝜒))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 662
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115
This theorem is referenced by: (None)
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