Theorem 3imtr4d

Description: More general version of 3imtr4i . Useful for converting conditional definitions in a formula. (Contributed by NM, 26-Oct-1995)

	Ref	Expression
Hypotheses	3imtr4d.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
	3imtr4d.2	$$⊢\phi \to \left(\theta \leftrightarrow \psi \right)$$
	3imtr4d.3	$$⊢\phi \to \left(\tau \leftrightarrow \chi \right)$$
Assertion	3imtr4d	$$⊢\phi \to \left(\theta \to \tau \right)$$

Proof

Step	Hyp	Ref	Expression
1		3imtr4d.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
2		3imtr4d.2	$$⊢\phi \to \left(\theta \leftrightarrow \psi \right)$$
3		3imtr4d.3	$$⊢\phi \to \left(\tau \leftrightarrow \chi \right)$$
4	1 3	sylibrd	$$⊢\phi \to \left(\psi \to \tau \right)$$
5	2 4	sylbid	$$⊢\phi \to \left(\theta \to \tau \right)$$