Theorem mpbidi 447

Description: A deduction from a biconditional, related to modus ponens.

Hypotheses

Ref	Expression
mpbidi.min	⊢ (θ → (φ → ψ))
mpbidi.maj	⊢ (φ → (ψ ↔ χ))

Assertion

Ref	Expression
mpbidi	⊢ (θ → (φ → χ))

Proof of Theorem mpbidi

Step	Hyp	Ref	Expression
1		mpbidi.min	. 2 ⊢ (θ → (φ → ψ))
2		mpbidi.maj	. . 3 ⊢ (φ → (ψ ↔ χ))
3	2	pm5.74i 443	. 2 ⊢ ((φ → ψ) ↔ (φ → χ))
4	1, 3	sylib 173	1 ⊢ (θ → (φ → χ))

Syntax hints:   → wi 2   ↔ wb 127

This theorem is referenced by:  tfrlem5 2953  pjss1co 5633

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-an 198

metamath.org