Theorem 3anbi3i 1194

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

Hypothesis

Ref	Expression
3anbi1i.1	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
3anbi3i	⊢ ((𝜒 ∧ 𝜃 ∧ 𝜑) ↔ (𝜒 ∧ 𝜃 ∧ 𝜓))

Proof of Theorem 3anbi3i

Step	Hyp	Ref	Expression
1		biid 171	. 2 ⊢ (𝜒 ↔ 𝜒)
2		biid 171	. 2 ⊢ (𝜃 ↔ 𝜃)
3		3anbi1i.1	. 2 ⊢ (𝜑 ↔ 𝜓)
4	1, 2, 3	3anbi123i 1190	1 ⊢ ((𝜒 ∧ 𝜃 ∧ 𝜑) ↔ (𝜒 ∧ 𝜃 ∧ 𝜓))

Syntax hints:   ↔ wb 105   ∧ w3a 980

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117  df-3an 982

This theorem is referenced by:  dfer2  6593  cbvsum  11525