Theorem e1bir

Description: Right biconditional form of e1a . sylibr is e1bir without virtual deductions. (Contributed by Alan Sare, 24-Jun-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	e1bir.1	$$⊢\left(\phi \to \psi \right)$$
	e1bir.2	$$⊢\chi \leftrightarrow \psi$$
Assertion	e1bir	$$⊢\left(\phi \to \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		e1bir.1	$$⊢\left(\phi \to \psi \right)$$
2		e1bir.2	$$⊢\chi \leftrightarrow \psi$$
3	2	biimpri	$$⊢\psi \to \chi$$
4	1 3	e1a	$$⊢\left(\phi \to \chi \right)$$