Theorem dedlema 1002

Description: Lemma for weak deduction theorem. See also ifptru 1023. (Contributed by NM, 26-Jun-2002.) (Proof shortened by Andrew Salmon, 7-May-2011.)

Assertion

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

Proof of Theorem dedlema

Step	Hyp	Ref	Expression
1		orc 400	. . 3 ⊢ ((𝜓 ∧ 𝜑) → ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑)))
2	1	expcom 451	. 2 ⊢ (𝜑 → (𝜓 → ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
3		simpl 473	. . . 4 ⊢ ((𝜓 ∧ 𝜑) → 𝜓)
4	3	a1i 11	. . 3 ⊢ (𝜑 → ((𝜓 ∧ 𝜑) → 𝜓))
5		pm2.24 121	. . . 4 ⊢ (𝜑 → (¬ 𝜑 → 𝜓))
6	5	adantld 483	. . 3 ⊢ (𝜑 → ((𝜒 ∧ ¬ 𝜑) → 𝜓))
7	4, 6	jaod 395	. 2 ⊢ (𝜑 → (((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑)) → 𝜓))
8	2, 7	impbid 202	1 ⊢ (𝜑 → (𝜓 ↔ ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))

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:  pm4.42  1004  elimhOLD  1033  dedtOLD  1034