Theorem 3anbi2i 1191

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 171	. 2 ⊢ (𝜒 ↔ 𝜒)
2		3anbi1i.1	. 2 ⊢ (𝜑 ↔ 𝜓)
3		biid 171	. 2 ⊢ (𝜃 ↔ 𝜃)
4	1, 2, 3	3anbi123i 1188	1 ⊢ ((𝜒 ∧ 𝜑 ∧ 𝜃) ↔ (𝜒 ∧ 𝜓 ∧ 𝜃))

Syntax hints:   ↔ wb 105   ∧ w3a 978

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 980

This theorem is referenced by:  seq3f1olemp  10505  seq3f1oleml  10506  fsum3  11398  issubg2m  13055