Theorem 3anbi1i 1157

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 1155	1 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) ↔ (𝜓 ∧ 𝜒 ∧ 𝜃))

Syntax hints:   ↔ wb 206   ∧ w3a 1086

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 207  df-an 396  df-3an 1088

This theorem is referenced by:  iinfi  9344  fzolb  13602  brfi1uzind  14449  opfi1uzind  14452  01sqrexlem5  15188  bitsmod  16382  isfunc  17806  txcn  23546  trfil2  23807  isclmp  25030  eulerpartlemn  34365  bnj976  34760  bnj543  34876  bnj594  34895  bnj917  34917  topdifinffinlem  37328  dath  39723  oeord2com  43293  ichexmpl1  47463  grtriproplem  47931  grtrif1o  47934  elfzolborelfzop1  48501  nnolog2flm1  48572  isthincd2  49419