Theorem 3imtr3g

Description: More general version of 3imtr3i . Useful for converting definitions in a formula. (Contributed by NM, 20-May-1996) (Proof shortened by Wolf Lammen, 20-Dec-2013)

	Ref	Expression
Hypotheses	3imtr3g.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
	3imtr3g.2	$$⊢\psi \leftrightarrow \theta$$
	3imtr3g.3	$$⊢\chi \leftrightarrow \tau$$
Assertion	3imtr3g	$$⊢\phi \to \left(\theta \to \tau \right)$$

Proof

Step	Hyp	Ref	Expression
1		3imtr3g.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
2		3imtr3g.2	$$⊢\psi \leftrightarrow \theta$$
3		3imtr3g.3	$$⊢\chi \leftrightarrow \tau$$
4	2 1	syl5bir	$$⊢\phi \to \left(\theta \to \chi \right)$$
5	4 3	syl6ib	$$⊢\phi \to \left(\theta \to \tau \right)$$