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Metamath Proof Explorer

Theorem mpbir3an

Description: Detach a conjunction of truths in a biconditional. (Contributed by NM, 16-Sep-2011)

Ref Expression
Hypotheses mpbir3an.1 ⊢ 𝜓
mpbir3an.2 ⊢ 𝜒
mpbir3an.3 ⊢ 𝜃
mpbir3an.4 ⊢ ( 𝜑 ↔ ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) )
Assertion mpbir3an 𝜑

Proof

Step Hyp Ref Expression
1 mpbir3an.1 ⊢ 𝜓
2 mpbir3an.2 ⊢ 𝜒
3 mpbir3an.3 ⊢ 𝜃
4 mpbir3an.4 ⊢ ( 𝜑 ↔ ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) )
5 1 2 3 3pm3.2i ⊢ ( 𝜓 ∧ 𝜒 ∧ 𝜃 )
6 5 4 mpbir ⊢ 𝜑