Theorem 3eqtr2i 2265

Description: An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.)

Hypotheses

Ref	Expression
3eqtr2i.1	⊢ 𝐴 = 𝐵
3eqtr2i.2	⊢ 𝐶 = 𝐵
3eqtr2i.3	⊢ 𝐶 = 𝐷

Assertion

Ref	Expression
3eqtr2i	⊢ 𝐴 = 𝐷

Proof of Theorem 3eqtr2i

Step	Hyp	Ref	Expression
1		3eqtr2i.1	. . 3 ⊢ 𝐴 = 𝐵
2		3eqtr2i.2	. . 3 ⊢ 𝐶 = 𝐵
3	1, 2	eqtr4i 2262	. 2 ⊢ 𝐴 = 𝐶
4		3eqtr2i.3	. 2 ⊢ 𝐶 = 𝐷
5	3, 4	eqtri 2259	1 ⊢ 𝐴 = 𝐷

This proof depends on syntax axioms:   = wceq 1402

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-4 1563  ax-17 1579  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-cleq 2231

This theorem is used by:  dfrab3  3509  iunid  4068  cnvcnv  5240  cocnvcnv2  5299  fmptap  5905  exmidfodomrlemim  7554  negdii  8612  halfpm6th  9530  numma  9830  numaddc  9834  6p5lem  9856  8p2e10  9866  binom2i  11100  0.999...  12307  flodddiv4  12722  6gcd4e2  12791  dfphi2  13021  mod2xnegi  13221  karatsuba  13233  1259lem1  13265  ballotfilem1  13272  ballotfilemfval0  13287  ballotfilemth  13333  cosq23lt0  16026  pigt3  16037  log2ublem3  16184  1sgm2ppw  16250  ppiqub  16254  bposlem8  16279  bposlem9  16280  2lgsoddprmlem3c  16394  2lgsoddprmlem3d  16395  nninfomni  17228