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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 632 . . 3 ((𝜑 ↔ (𝜓𝜒)) → ((𝜑 ∧ ¬ 𝜓) ↔ ((𝜓𝜒) ∧ ¬ 𝜓)))
2 biorf 934 . . . . 5 𝜓 → (𝜒 ↔ (𝜓𝜒)))
32bicomd 222 . . . 4 𝜓 → ((𝜓𝜒) ↔ 𝜒))
43pm5.32ri 576 . . 3 (((𝜓𝜒) ∧ ¬ 𝜓) ↔ (𝜒 ∧ ¬ 𝜓))
51, 4bitrdi 287 . 2 ((𝜑 ↔ (𝜓𝜒)) → ((𝜑 ∧ ¬ 𝜓) ↔ (𝜒 ∧ ¬ 𝜓)))
6 abai 824 . . 3 ((𝜒 ∧ ¬ 𝜓) ↔ (𝜒 ∧ (𝜒 → ¬ 𝜓)))
76rbaib 539 . 2 ((𝜒 → ¬ 𝜓) → ((𝜒 ∧ ¬ 𝜓) ↔ 𝜒))
85, 7sylan9bbr 511 1 (((𝜒 → ¬ 𝜓) ∧ (𝜑 ↔ (𝜓𝜒))) → ((𝜑 ∧ ¬ 𝜓) ↔ 𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  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 206  df-an 397  df-or 845
This theorem is referenced by: (None)
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