Theorem ibd 167

Description: Deduction that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 26-Jun-2004.)

Hypothesis

Ref	Expression
ibd.1	⊢ (𝜑 → (𝜓 → (𝜓 ↔ 𝜒)))

Assertion

Ref	Expression
ibd	⊢ (𝜑 → (𝜓 → 𝜒))

Proof of Theorem ibd

Step	Hyp	Ref	Expression
1		ibd.1	. 2 ⊢ (𝜑 → (𝜓 → (𝜓 ↔ 𝜒)))
2		bi1 111	. 2 ⊢ ((𝜓 ↔ 𝜒) → (𝜓 → 𝜒))
3	1, 2	syli 33	1 ⊢ (𝜑 → (𝜓 → 𝜒))

Syntax hints:   → wi 4   ↔ wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  pm5.21ndd  621  oibabs  800  sssnm  3525