Theorem bitrd

Description: Deduction form of bitri . (Contributed by NM, 12-Mar-1993) (Proof shortened by Wolf Lammen, 14-Apr-2013)

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

Proof

Step	Hyp	Ref	Expression
1		bitrd.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2		bitrd.2	$$⊢\phi \to \left(\chi \leftrightarrow \theta \right)$$
3	1	pm5.74i	$$⊢\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \chi \right)$$
4	2	pm5.74i	$$⊢\left(\phi \to \chi \right)\leftrightarrow \left(\phi \to \theta \right)$$
5	3 4	bitri	$$⊢\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \theta \right)$$
6	5	pm5.74ri	$$⊢\phi \to \left(\psi \leftrightarrow \theta \right)$$