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Metamath Proof Explorer

Theorem impbid

Description: Deduce an equivalence from two implications. Deduction associated with impbi and impbii . (Contributed by NM, 24-Jan-1993) Prove it from impbid21d . (Revised by Wolf Lammen, 3-Nov-2012)

	Ref	Expression
Hypotheses	impbid.1	$⊢ φ \to (ψ \to χ)$
	impbid.2	$⊢ φ \to (χ \to ψ)$
Assertion	impbid	$⊢ φ \to (ψ \leftrightarrow χ)$

Proof

Step	Hyp	Ref	Expression
1		impbid.1	$⊢ φ \to (ψ \to χ)$
2		impbid.2	$⊢ φ \to (χ \to ψ)$
3	1 2	impbid21d	$⊢ φ \to (φ \to (ψ \leftrightarrow χ))$
4	3	pm2.43i	$⊢ φ \to (ψ \leftrightarrow χ)$