Theorem elimh 571

Description: Hypothesis builder for weak deduction theorem. For more information, see the Deduction Theorem link on the Metamath Proof Explorer home page.

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 569	. . . 4 ⊢ (χ → (φ ↔ ((φ ∧ χ) ∨ (ψ ∧ ¬ χ))))
2		elimh.1	. . . 4 ⊢ ((φ ↔ ((φ ∧ χ) ∨ (ψ ∧ ¬ χ))) → (χ ↔ τ))
3	1, 2	syl 12	. . 3 ⊢ (χ → (χ ↔ τ))
4	3	ibi 449	. 2 ⊢ (χ → τ)
5		elimh.3	. . 3 ⊢ θ
6		dedlemb 570	. . . 4 ⊢ (¬ χ → (ψ ↔ ((φ ∧ χ) ∨ (ψ ∧ ¬ χ))))
7		elimh.2	. . . 4 ⊢ ((ψ ↔ ((φ ∧ χ) ∨ (ψ ∧ ¬ χ))) → (θ ↔ τ))
8	6, 7	syl 12	. . 3 ⊢ (¬ χ → (θ ↔ τ))
9	5, 8	mpbii 168	. 2 ⊢ (¬ χ → τ)
10	4, 9	pm2.61i 110	1 ⊢ τ

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196

This theorem is referenced by:  con3th 573

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198

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