Theorem 3imtr3d

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

	Ref	Expression
Hypotheses	3imtr3d.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
	3imtr3d.2	$$⊢\phi \to \left(\psi \leftrightarrow \theta \right)$$
	3imtr3d.3	$$⊢\phi \to \left(\chi \leftrightarrow \tau \right)$$
Assertion	3imtr3d	$$⊢\phi \to \left(\theta \to \tau \right)$$

Proof

Step	Hyp	Ref	Expression
1		3imtr3d.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
2		3imtr3d.2	$$⊢\phi \to \left(\psi \leftrightarrow \theta \right)$$
3		3imtr3d.3	$$⊢\phi \to \left(\chi \leftrightarrow \tau \right)$$
4	1 3	sylibd	$$⊢\phi \to \left(\psi \to \tau \right)$$
5	2 4	sylbird	$$⊢\phi \to \left(\theta \to \tau \right)$$