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Theorem pm5.75 1026
 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.) (Proof shortened by Kyle Wyonch, 12-Feb-2021.)
Assertion
Ref Expression
pm5.75 (((𝜒 → ¬ 𝜓) ∧ (𝜑 ↔ (𝜓𝜒))) → ((𝜑 ∧ ¬ 𝜓) ↔ 𝜒))

Proof of Theorem pm5.75
StepHypRef Expression
1 anbi1 634 . . 3 ((𝜑 ↔ (𝜓𝜒)) → ((𝜑 ∧ ¬ 𝜓) ↔ ((𝜓𝜒) ∧ ¬ 𝜓)))
2 biorf 934 . . . . 5 𝜓 → (𝜒 ↔ (𝜓𝜒)))
32bicomd 226 . . . 4 𝜓 → ((𝜓𝜒) ↔ 𝜒))
43pm5.32ri 579 . . 3 (((𝜓𝜒) ∧ ¬ 𝜓) ↔ (𝜒 ∧ ¬ 𝜓))
51, 4syl6bb 290 . 2 ((𝜑 ↔ (𝜓𝜒)) → ((𝜑 ∧ ¬ 𝜓) ↔ (𝜒 ∧ ¬ 𝜓)))
6 abai 825 . . 3 ((𝜒 ∧ ¬ 𝜓) ↔ (𝜒 ∧ (𝜒 → ¬ 𝜓)))
76rbaib 542 . 2 ((𝜒 → ¬ 𝜓) → ((𝜒 ∧ ¬ 𝜓) ↔ 𝜒))
85, 7sylan9bbr 514 1 (((𝜒 → ¬ 𝜓) ∧ (𝜑 ↔ (𝜓𝜒))) → ((𝜑 ∧ ¬ 𝜓) ↔ 𝜒))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844 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 210  df-an 400  df-or 845 This theorem is referenced by: (None)
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