Theorem 3anbi1i 1187

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 1185	1 |- ( ( ph /\ ch /\ th ) <-> ( ps /\ ch /\ th ) )

Syntax hints:   <-> wb 184   /\ w3a 973

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 975

This theorem is referenced by:  iinfi  7895  fzolb  11832  sqrlem5  13092  bitsmod  14098  isfunc  15280  istps5OLD  19552  txcn  20253  trfil2  20514  eulerpartlemn  28517  bnj976  33979  bnj543  34094  bnj594  34113  bnj917  34135  dath  35603