Theorem 3anbi2i 1176

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
3anbi2i	⊢ ((𝜒 ∧ 𝜑 ∧ 𝜃) ↔ (𝜒 ∧ 𝜓 ∧ 𝜃))

Proof of Theorem 3anbi2i

Step	Hyp	Ref	Expression
1		biid 264	. 2 ⊢ (𝜒 ↔ 𝜒)
2		3anbi1i.1	. 2 ⊢ (𝜑 ↔ 𝜓)
3		biid 264	. 2 ⊢ (𝜃 ↔ 𝜃)
4	1, 2, 3	3anbi123i 1173	1 ⊢ ((𝜒 ∧ 𝜑 ∧ 𝜃) ↔ (𝜒 ∧ 𝜓 ∧ 𝜃))

This proof depends on syntax axioms:   ↔ wb 209   ∧ w3a 1103

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105

This theorem is used by:  f13dfv  7282  axgroth4  10917  fi1uzind  14652  bnj543  35523  bnj916  35563  topdifinffinlem  38271