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Metamath Proof Explorer

Theorem 3anbi3i

Description: Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006)

Ref Expression
Hypothesis 3anbi1i.1 ⊢ ( 𝜑 ↔ 𝜓 )
Assertion 3anbi3i ( ( 𝜒 ∧ 𝜃 ∧ 𝜑 ) ↔ ( 𝜒 ∧ 𝜃 ∧ 𝜓 ) )

Proof

Step Hyp Ref Expression
1 3anbi1i.1 ⊢ ( 𝜑 ↔ 𝜓 )
2 biid ⊢ ( 𝜒 ↔ 𝜒 )
3 biid ⊢ ( 𝜃 ↔ 𝜃 )
4 2 3 1 3anbi123i ⊢ ( ( 𝜒 ∧ 𝜃 ∧ 𝜑 ) ↔ ( 𝜒 ∧ 𝜃 ∧ 𝜓 ) )