Metamath Proof Explorer


Theorem 3anbi13d

Description: Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006)

Ref Expression
Hypotheses 3anbi12d.1 ( 𝜑 → ( 𝜓𝜒 ) )
3anbi12d.2 ( 𝜑 → ( 𝜃𝜏 ) )
Assertion 3anbi13d ( 𝜑 → ( ( 𝜓𝜂𝜃 ) ↔ ( 𝜒𝜂𝜏 ) ) )

Proof

Step Hyp Ref Expression
1 3anbi12d.1 ( 𝜑 → ( 𝜓𝜒 ) )
2 3anbi12d.2 ( 𝜑 → ( 𝜃𝜏 ) )
3 biidd ( 𝜑 → ( 𝜂𝜂 ) )
4 1 3 2 3anbi123d ( 𝜑 → ( ( 𝜓𝜂𝜃 ) ↔ ( 𝜒𝜂𝜏 ) ) )