Theorem 3bitr4rd 277

Description: Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)

Hypotheses

Ref	Expression
3bitr4d.1	⊢ (φ → (ψ ↔ χ))
3bitr4d.2	⊢ (φ → (θ ↔ ψ))
3bitr4d.3	⊢ (φ → (τ ↔ χ))

Assertion

Ref	Expression
3bitr4rd	⊢ (φ → (τ ↔ θ))

Proof of Theorem 3bitr4rd

Step	Hyp	Ref	Expression
1		3bitr4d.3	. . 3 ⊢ (φ → (τ ↔ χ))
2		3bitr4d.1	. . 3 ⊢ (φ → (ψ ↔ χ))
3	1, 2	bitr4d 247	. 2 ⊢ (φ → (τ ↔ ψ))
4		3bitr4d.2	. 2 ⊢ (φ → (θ ↔ ψ))
5	3, 4	bitr4d 247	1 ⊢ (φ → (τ ↔ θ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176

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

This theorem is used by: (None)