Theorem pm5.75OLD 979

Description: Obsolete proof of pm5.75 978 as of 12-Feb-2021. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 23-Dec-2012.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
pm5.75OLD	⊢ (((𝜒 → ¬ 𝜓) ∧ (𝜑 ↔ (𝜓 ∨ 𝜒))) → ((𝜑 ∧ ¬ 𝜓) ↔ 𝜒))

Proof of Theorem pm5.75OLD

Step	Hyp	Ref	Expression
1		anbi1 743	. . 3 ⊢ ((𝜑 ↔ (𝜓 ∨ 𝜒)) → ((𝜑 ∧ ¬ 𝜓) ↔ ((𝜓 ∨ 𝜒) ∧ ¬ 𝜓)))
2		orcom 402	. . . . 5 ⊢ ((𝜓 ∨ 𝜒) ↔ (𝜒 ∨ 𝜓))
3	2	anbi1i 731	. . . 4 ⊢ (((𝜓 ∨ 𝜒) ∧ ¬ 𝜓) ↔ ((𝜒 ∨ 𝜓) ∧ ¬ 𝜓))
4		pm5.61 749	. . . 4 ⊢ (((𝜒 ∨ 𝜓) ∧ ¬ 𝜓) ↔ (𝜒 ∧ ¬ 𝜓))
5	3, 4	bitri 264	. . 3 ⊢ (((𝜓 ∨ 𝜒) ∧ ¬ 𝜓) ↔ (𝜒 ∧ ¬ 𝜓))
6	1, 5	syl6bb 276	. 2 ⊢ ((𝜑 ↔ (𝜓 ∨ 𝜒)) → ((𝜑 ∧ ¬ 𝜓) ↔ (𝜒 ∧ ¬ 𝜓)))
7		pm4.71 662	. . . 4 ⊢ ((𝜒 → ¬ 𝜓) ↔ (𝜒 ↔ (𝜒 ∧ ¬ 𝜓)))
8	7	biimpi 206	. . 3 ⊢ ((𝜒 → ¬ 𝜓) → (𝜒 ↔ (𝜒 ∧ ¬ 𝜓)))
9	8	bicomd 213	. 2 ⊢ ((𝜒 → ¬ 𝜓) → ((𝜒 ∧ ¬ 𝜓) ↔ 𝜒))
10	6, 9	sylan9bbr 737	1 ⊢ (((𝜒 → ¬ 𝜓) ∧ (𝜑 ↔ (𝜓 ∨ 𝜒))) → ((𝜑 ∧ ¬ 𝜓) ↔ 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384

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 197  df-or 385  df-an 386

This theorem is referenced by: (None)