Theorem elimhOLD 1033

Description: Old version of elimh 1030. Obsolete as of 16-Mar-2021. (Contributed by NM, 26-Jun-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
elimhOLD.1	⊢ ((𝜑 ↔ ((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ ¬ 𝜒))) → (𝜒 ↔ 𝜏))
elimhOLD.2	⊢ ((𝜓 ↔ ((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ ¬ 𝜒))) → (𝜃 ↔ 𝜏))
elimhOLD.3	⊢ 𝜃

Assertion

Ref	Expression
elimhOLD	⊢ 𝜏

Proof of Theorem elimhOLD

Step	Hyp	Ref	Expression
1		dedlema 1002	. . . 4 ⊢ (𝜒 → (𝜑 ↔ ((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ ¬ 𝜒))))
2		elimhOLD.1	. . . 4 ⊢ ((𝜑 ↔ ((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ ¬ 𝜒))) → (𝜒 ↔ 𝜏))
3	1, 2	syl 17	. . 3 ⊢ (𝜒 → (𝜒 ↔ 𝜏))
4	3	ibi 256	. 2 ⊢ (𝜒 → 𝜏)
5		elimhOLD.3	. . 3 ⊢ 𝜃
6		dedlemb 1003	. . . 4 ⊢ (¬ 𝜒 → (𝜓 ↔ ((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ ¬ 𝜒))))
7		elimhOLD.2	. . . 4 ⊢ ((𝜓 ↔ ((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ ¬ 𝜒))) → (𝜃 ↔ 𝜏))
8	6, 7	syl 17	. . 3 ⊢ (¬ 𝜒 → (𝜃 ↔ 𝜏))
9	5, 8	mpbii 223	. 2 ⊢ (¬ 𝜒 → 𝜏)
10	4, 9	pm2.61i 176	1 ⊢ 𝜏

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:  con3OLD  1035