Theorem 3bitr3d 274

Description: Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula. (Contributed by NM, 24-Apr-1996.)

Hypotheses

Ref	Expression
3bitr3d.1	⊢ (φ → (ψ ↔ χ))
3bitr3d.2	⊢ (φ → (ψ ↔ θ))
3bitr3d.3	⊢ (φ → (χ ↔ τ))

Assertion

Ref	Expression
3bitr3d	⊢ (φ → (θ ↔ τ))

Proof of Theorem 3bitr3d

Step	Hyp	Ref	Expression
1		3bitr3d.2	. . 3 ⊢ (φ → (ψ ↔ θ))
2		3bitr3d.1	. . 3 ⊢ (φ → (ψ ↔ χ))
3	1, 2	bitr3d 246	. 2 ⊢ (φ → (θ ↔ χ))
4		3bitr3d.3	. 2 ⊢ (φ → (χ ↔ τ))
5	3, 4	bitrd 244	1 ⊢ (φ → (θ ↔ τ))

Syntax hints:   → wi 4   ↔ wb 176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177

This theorem is referenced by:  sbcne12g  3155  csbcomg  3160  eloprabga  5579  ereldm  5972