Theorem dedt 572

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

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 569	. 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 169	. 2 ⊢ ((φ ↔ ((φ ∧ χ) ∨ (ψ ∧ ¬ χ))) → θ)
5	1, 4	syl 12	1 ⊢ (χ → θ)

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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