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Metamath Proof Explorer

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	$⊢ (φ \to ψ)$
	e1bir.2	$⊢ χ \leftrightarrow ψ$
Assertion	e1bir	$⊢ (φ \to χ)$

Proof

Step	Hyp	Ref	Expression
1		e1bir.1	$⊢ (φ \to ψ)$
2		e1bir.2	$⊢ χ \leftrightarrow ψ$
3	2	biimpri	$⊢ ψ \to χ$
4	1 3	e1a	$⊢ (φ \to χ)$