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 ↔ (𝜓 ∨ 𝜒))) → ((𝜑 ∧ ¬ 𝜓) ↔ 𝜒))

Proof of Theorem pm5.75

Step	Hyp	Ref	Expression
1		anbi1 632	. . 3 ⊢ ((𝜑 ↔ (𝜓 ∨ 𝜒)) → ((𝜑 ∧ ¬ 𝜓) ↔ ((𝜓 ∨ 𝜒) ∧ ¬ 𝜓)))
2		biorf 934	. . . . 5 ⊢ (¬ 𝜓 → (𝜒 ↔ (𝜓 ∨ 𝜒)))
3	2	bicomd 222	. . . 4 ⊢ (¬ 𝜓 → ((𝜓 ∨ 𝜒) ↔ 𝜒))
4	3	pm5.32ri 576	. . 3 ⊢ (((𝜓 ∨ 𝜒) ∧ ¬ 𝜓) ↔ (𝜒 ∧ ¬ 𝜓))
5	1, 4	bitrdi 287	. 2 ⊢ ((𝜑 ↔ (𝜓 ∨ 𝜒)) → ((𝜑 ∧ ¬ 𝜓) ↔ (𝜒 ∧ ¬ 𝜓)))
6		abai 824	. . 3 ⊢ ((𝜒 ∧ ¬ 𝜓) ↔ (𝜒 ∧ (𝜒 → ¬ 𝜓)))
7	6	rbaib 539	. 2 ⊢ ((𝜒 → ¬ 𝜓) → ((𝜒 ∧ ¬ 𝜓) ↔ 𝜒))
8	5, 7	sylan9bbr 511	1 ⊢ (((𝜒 → ¬ 𝜓) ∧ (𝜑 ↔ (𝜓 ∨ 𝜒))) → ((𝜑 ∧ ¬ 𝜓) ↔ 𝜒))

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)