Theorem 3anbi2i 1154

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 263	. 2 ⊢ (𝜒 ↔ 𝜒)
2		3anbi1i.1	. 2 ⊢ (𝜑 ↔ 𝜓)
3		biid 263	. 2 ⊢ (𝜃 ↔ 𝜃)
4	1, 2, 3	3anbi123i 1151	1 ⊢ ((𝜒 ∧ 𝜑 ∧ 𝜃) ↔ (𝜒 ∧ 𝜓 ∧ 𝜃))

Syntax hints:   ↔ wb 208   ∧ w3a 1083

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 209  df-an 399  df-3an 1085

This theorem is referenced by:  f13dfv  7033  axgroth4  10256  fi1uzind  13858  bnj543  32167  bnj916  32207  topdifinffinlem  34630