Theorem dedt 923

Description: The 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
dedt.1	⊢ ((φ ↔ ((φ ∧ χ) ∨ (ψ ∧ ¬ χ))) → (θ ↔ τ))
dedt.2	⊢ τ

Assertion

Ref	Expression
dedt	⊢ (χ → θ)

Proof of Theorem dedt

Step	Hyp	Ref	Expression
1		dedlema 920	. 2 ⊢ (χ → (φ ↔ ((φ ∧ χ) ∨ (ψ ∧ ¬ χ))))
2		dedt.2	. . 3 ⊢ τ
3		dedt.1	. . 3 ⊢ ((φ ↔ ((φ ∧ χ) ∨ (ψ ∧ ¬ χ))) → (θ ↔ τ))
4	2, 3	mpbiri 224	. 2 ⊢ ((φ ↔ ((φ ∧ χ) ∨ (ψ ∧ ¬ χ))) → θ)
5	1, 4	syl 15	1 ⊢ (χ → θ)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358

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

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360

This theorem is used by:  con3th  924