Theorem 3bitrd 270

Description: Deduction from transitivity of biconditional. (Contributed by NM, 13-Aug-1999.)

Hypotheses

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

Assertion

Ref	Expression
3bitrd	⊢ (φ → (ψ ↔ τ))

Proof of Theorem 3bitrd

Step	Hyp	Ref	Expression
1		3bitrd.1	. . 3 ⊢ (φ → (ψ ↔ χ))
2		3bitrd.2	. . 3 ⊢ (φ → (χ ↔ θ))
3	1, 2	bitrd 244	. 2 ⊢ (φ → (ψ ↔ θ))
4		3bitrd.3	. 2 ⊢ (φ → (θ ↔ τ))
5	3, 4	bitrd 244	1 ⊢ (φ → (ψ ↔ τ))

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:  sbceqal  3098  sbcnel12g  3154  elimhyp3v  3713  elimhyp4v  3714  keephyp3v  3719  opkelopkabg  4246  dfphi2  4570