Theorem 3anbi1i 1179

Description: Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)

Hypothesis

Ref	Expression
3anbi1i.1	|- ( ph <-> ps )

Assertion

Ref	Expression
3anbi1i	|- ( ( ph /\ ch /\ th ) <-> ( ps /\ ch /\ th ) )

Proof of Theorem 3anbi1i

Step	Hyp	Ref	Expression
1		3anbi1i.1	. 2 |- ( ph <-> ps )
2		biid 236	. 2 |- ( ch <-> ch )
3		biid 236	. 2 |- ( th <-> th )
4	1, 2, 3	3anbi123i 1177	1 |- ( ( ph /\ ch /\ th ) <-> ( ps /\ ch /\ th ) )

Syntax hints:   <-> wb 184   /\ w3a 965

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 185  df-an 371  df-3an 967

This theorem is referenced by:  iinfi  7777  fzolb  11674  sqrlem5  12853  bitsmod  13749  isfunc  14892  istps5OLD  18660  txcn  19330  trfil2  19591  eulerpartlemn  26907  bnj976  32088  bnj543  32203  bnj594  32222  bnj917  32244  dath  33703