Metamath Proof Explorer

Theorem pm4.71

Description: Implication in terms of biconditional and conjunction. Theorem *4.71 of WhiteheadRussell p. 120. (Contributed by NM, 21-Jun-1993) (Proof shortened by Wolf Lammen, 2-Dec-2012)

		Ref	Expression
	Assertion	pm4.71	$⊢ (φ \to ψ) \leftrightarrow (φ \leftrightarrow (φ \land ψ))$

Proof

Step	Hyp	Ref	Expression
1		simpl	$⊢ (φ \land ψ) \to φ$
2	1	biantru	$⊢ (φ \to (φ \land ψ)) \leftrightarrow ((φ \to (φ \land ψ)) \land ((φ \land ψ) \to φ))$
3		anclb	$⊢ (φ \to ψ) \leftrightarrow (φ \to (φ \land ψ))$
4		dfbi2	$⊢ (φ \leftrightarrow (φ \land ψ)) \leftrightarrow ((φ \to (φ \land ψ)) \land ((φ \land ψ) \to φ))$
5	2 3 4	3bitr4i	$⊢ (φ \to ψ) \leftrightarrow (φ \leftrightarrow (φ \land ψ))$