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Metamath Proof Explorer

Theorem mpbir3and

Description: Detach a conjunction of truths in a biconditional. (Contributed by Mario Carneiro, 11-May-2014) (Revised by Mario Carneiro, 9-Jan-2015)

	Ref	Expression
Hypotheses	mpbir3and.1	$⊢ φ \to χ$
	mpbir3and.2	$⊢ φ \to θ$
	mpbir3and.3	$⊢ φ \to τ$
	mpbir3and.4	$⊢ φ \to (ψ \leftrightarrow (χ \land θ \land τ))$
Assertion	mpbir3and	$⊢ φ \to ψ$

Proof

Step	Hyp	Ref	Expression
1		mpbir3and.1	$⊢ φ \to χ$
2		mpbir3and.2	$⊢ φ \to θ$
3		mpbir3and.3	$⊢ φ \to τ$
4		mpbir3and.4	$⊢ φ \to (ψ \leftrightarrow (χ \land θ \land τ))$
5	1 2 3	3jca	$⊢ φ \to (χ \land θ \land τ)$
6	5 4	mpbird	$⊢ φ \to ψ$