Theorem dedlemb 911

Description: Lemma for iffalse 3359. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 7-May-2011.)

Assertion

Ref	Expression
dedlemb	⊢ (¬ 𝜑 → (𝜒 ↔ ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))

Proof of Theorem dedlemb

Step	Hyp	Ref	Expression
1		olc 664	. . 3 ⊢ ((𝜒 ∧ ¬ 𝜑) → ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑)))
2	1	expcom 114	. 2 ⊢ (¬ 𝜑 → (𝜒 → ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
3		pm2.21 579	. . . 4 ⊢ (¬ 𝜑 → (𝜑 → 𝜒))
4	3	adantld 272	. . 3 ⊢ (¬ 𝜑 → ((𝜓 ∧ 𝜑) → 𝜒))
5		simpl 107	. . . 4 ⊢ ((𝜒 ∧ ¬ 𝜑) → 𝜒)
6	5	a1i 9	. . 3 ⊢ (¬ 𝜑 → ((𝜒 ∧ ¬ 𝜑) → 𝜒))
7	4, 6	jaod 669	. 2 ⊢ (¬ 𝜑 → (((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑)) → 𝜒))
8	2, 7	impbid 127	1 ⊢ (¬ 𝜑 → (𝜒 ↔ ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103   ∨ wo 661

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 577  ax-io 662

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  iffalse  3359