Theorem 3imtr3d 258

Description: More general version of 3imtr3i 256. Useful for converting conditional definitions in a formula. (Contributed by NM, 8-Apr-1996.)

Hypotheses

Ref	Expression
3imtr3d.1	⊢ (φ → (ψ → χ))
3imtr3d.2	⊢ (φ → (ψ ↔ θ))
3imtr3d.3	⊢ (φ → (χ ↔ τ))

Assertion

Ref	Expression
3imtr3d	⊢ (φ → (θ → τ))

Proof of Theorem 3imtr3d

Step	Hyp	Ref	Expression
1		3imtr3d.2	. 2 ⊢ (φ → (ψ ↔ θ))
2		3imtr3d.1	. . 3 ⊢ (φ → (ψ → χ))
3		3imtr3d.3	. . 3 ⊢ (φ → (χ ↔ τ))
4	2, 3	sylibd 205	. 2 ⊢ (φ → (ψ → τ))
5	1, 4	sylbird 226	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: (None)