Theorem dedlemb 1042

Description: Lemma for weak deduction theorem. See also ifpfal 1072. (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 865	. . 3 ⊢ ((𝜒 ∧ ¬ 𝜑) → ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑)))
2	1	expcom 417	. 2 ⊢ (¬ 𝜑 → (𝜒 → ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
3		pm2.21 123	. . . 4 ⊢ (¬ 𝜑 → (𝜑 → 𝜒))
4	3	adantld 494	. . 3 ⊢ (¬ 𝜑 → ((𝜓 ∧ 𝜑) → 𝜒))
5		simpl 486	. . . 4 ⊢ ((𝜒 ∧ ¬ 𝜑) → 𝜒)
6	5	a1i 11	. . 3 ⊢ (¬ 𝜑 → ((𝜒 ∧ ¬ 𝜑) → 𝜒))
7	4, 6	jaod 856	. 2 ⊢ (¬ 𝜑 → (((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑)) → 𝜒))
8	2, 7	impbid 215	1 ⊢ (¬ 𝜑 → (𝜒 ↔ ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))

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:  cases2  1043  pm4.42  1049  iffalse  4434