Theorem ibd 177

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		biimp 117	. 2 ⊢ ((𝜓 ↔ 𝜒) → (𝜓 → 𝜒))
3	1, 2	syli 37	1 ⊢ (𝜑 → (𝜓 → 𝜒))

Syntax hints:   → wi 4   ↔ wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  pm5.21ndd  695  oibabs  704  sssnm  3734