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  9368  fzolb  13626  brfi1uzind  14473  opfi1uzind  14476  01sqrexlem5  15212  bitsmod  16406  isfunc  17826  txcn  23513  trfil2  23774  isclmp  24997  eulerpartlemn  34372  bnj976  34767  bnj543  34883  bnj594  34902  bnj917  34924  topdifinffinlem  37335  dath  39730  oeord2com  43300  ichexmpl1  47470  grtriproplem  47938  grtrif1o  47941  elfzolborelfzop1  48508  nnolog2flm1  48579  isthincd2  49426