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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 → 𝜒 )
	mpbir3and.2	⊢ ( 𝜑 → 𝜃 )
	mpbir3and.3	⊢ ( 𝜑 → 𝜏 )
	mpbir3and.4	⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜒 ∧ 𝜃 ∧ 𝜏 ) ) )
Assertion	mpbir3and	⊢ ( 𝜑 → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		mpbir3and.1	⊢ ( 𝜑 → 𝜒 )
2		mpbir3and.2	⊢ ( 𝜑 → 𝜃 )
3		mpbir3and.3	⊢ ( 𝜑 → 𝜏 )
4		mpbir3and.4	⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜒 ∧ 𝜃 ∧ 𝜏 ) ) )
5	1 2 3	3jca	⊢ ( 𝜑 → ( 𝜒 ∧ 𝜃 ∧ 𝜏 ) )
6	5 4	mpbird	⊢ ( 𝜑 → 𝜓 )