Theorem elimh 922

Description: Hypothesis builder for weak deduction theorem. For more information, see the Deduction Theorem link on the Metamath Proof Explorer home page. (Contributed by NM, 26-Jun-2002.)

Hypotheses

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

Assertion

Ref	Expression
elimh	⊢ τ

Proof of Theorem elimh

Step	Hyp	Ref	Expression
1		dedlema 920	. . . 4 ⊢ (χ → (φ ↔ ((φ ∧ χ) ∨ (ψ ∧ ¬ χ))))
2		elimh.1	. . . 4 ⊢ ((φ ↔ ((φ ∧ χ) ∨ (ψ ∧ ¬ χ))) → (χ ↔ τ))
3	1, 2	syl 15	. . 3 ⊢ (χ → (χ ↔ τ))
4	3	ibi 232	. 2 ⊢ (χ → τ)
5		elimh.3	. . 3 ⊢ θ
6		dedlemb 921	. . . 4 ⊢ (¬ χ → (ψ ↔ ((φ ∧ χ) ∨ (ψ ∧ ¬ χ))))
7		elimh.2	. . . 4 ⊢ ((ψ ↔ ((φ ∧ χ) ∨ (ψ ∧ ¬ χ))) → (θ ↔ τ))
8	6, 7	syl 15	. . 3 ⊢ (¬ χ → (θ ↔ τ))
9	5, 8	mpbii 202	. 2 ⊢ (¬ χ → τ)
10	4, 9	pm2.61i 156	1 ⊢ τ

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358

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 177  df-or 359  df-an 360

This theorem is referenced by:  con3th  924