Theorem dedlemb 1062

Description: Lemma for weak deduction theorem. See also ifpfal 1092. (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 882	. . 3 ⊢ ((𝜒 ∧ ¬ 𝜑) → ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑)))
2	1	expcom 419	. 2 ⊢ (¬ 𝜑 → (𝜒 → ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
3		pm2.21 124	. . . 4 ⊢ (¬ 𝜑 → (𝜑 → 𝜒))
4	3	adantld 496	. . 3 ⊢ (¬ 𝜑 → ((𝜓 ∧ 𝜑) → 𝜒))
5		simpl 488	. . . 4 ⊢ ((𝜒 ∧ ¬ 𝜑) → 𝜒)
6	5	a1i 11	. . 3 ⊢ (¬ 𝜑 → ((𝜒 ∧ ¬ 𝜑) → 𝜒))
7	4, 6	jaod 873	. 2 ⊢ (¬ 𝜑 → (((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑)) → 𝜒))
8	2, 7	impbid 215	1 ⊢ (¬ 𝜑 → (𝜒 ↔ ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862

This theorem is used by:  cases2  1063  pm4.42  1069  iffalse  4491