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Metamath Proof Explorer

Theorem 3anbi2d

Description: Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006)

Ref Expression
Hypothesis 3anbi1d.1 ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
Assertion 3anbi2d ( 𝜑 → ( ( 𝜃 ∧ 𝜓 ∧ 𝜏 ) ↔ ( 𝜃 ∧ 𝜒 ∧ 𝜏 ) ) )

Proof

Step Hyp Ref Expression
1 3anbi1d.1 ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2 biidd ⊢ ( 𝜑 → ( 𝜃 ↔ 𝜃 ) )
3 2 1 3anbi12d ⊢ ( 𝜑 → ( ( 𝜃 ∧ 𝜓 ∧ 𝜏 ) ↔ ( 𝜃 ∧ 𝜒 ∧ 𝜏 ) ) )