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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 → ( 𝜓 ↔ 𝜒 ) )
3anbi12d.2 ⊢ ( 𝜑 → ( 𝜃 ↔ 𝜏 ) )
Assertion 3anbi13d ( 𝜑 → ( ( 𝜓 ∧ 𝜂 ∧ 𝜃 ) ↔ ( 𝜒 ∧ 𝜂 ∧ 𝜏 ) ) )

Proof

Step Hyp Ref Expression
1 3anbi12d.1 ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2 3anbi12d.2 ⊢ ( 𝜑 → ( 𝜃 ↔ 𝜏 ) )
3 biidd ⊢ ( 𝜑 → ( 𝜂 ↔ 𝜂 ) )
4 1 3 2 3anbi123d ⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜂 ∧ 𝜃 ) ↔ ( 𝜒 ∧ 𝜂 ∧ 𝜏 ) ) )