Theorem 3anbi1i 1158

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
3anbi1i	⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) ↔ (𝜓 ∧ 𝜒 ∧ 𝜃))

Proof of Theorem 3anbi1i

Step	Hyp	Ref	Expression
1		3anbi1i.1	. 2 ⊢ (𝜑 ↔ 𝜓)
2		biid 261	. 2 ⊢ (𝜒 ↔ 𝜒)
3		biid 261	. 2 ⊢ (𝜃 ↔ 𝜃)
4	1, 2, 3	3anbi123i 1156	1 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) ↔ (𝜓 ∧ 𝜒 ∧ 𝜃))

Syntax hints:   ↔ wb 205   ∧ w3a 1088

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 206  df-an 398  df-3an 1090

This theorem is referenced by:  iinfi  9412  fzolb  13638  brfi1uzind  14459  opfi1uzind  14462  01sqrexlem5  15193  bitsmod  16377  isfunc  17814  txcn  23130  trfil2  23391  isclmp  24613  eulerpartlemn  33411  bnj976  33819  bnj543  33935  bnj594  33954  bnj917  33976  topdifinffinlem  36276  dath  38655  oeord2com  42109  ichexmpl1  46185  elfzolborelfzop1  47248  nnolog2flm1  47324  isthincd2  47706