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  7553  negdii  8611  halfpm6th  9529  numma  9829  numaddc  9833  6p5lem  9855  8p2e10  9865  binom2i  11098  0.999...  12304  flodddiv4  12719  6gcd4e2  12788  dfphi2  13018  mod2xnegi  13218  karatsuba  13230  1259lem1  13262  ballotfilem1  13269  ballotfilemfval0  13284  ballotfilemth  13330  cosq23lt0  15984  pigt3  15995  log2ublem3  16142  1sgm2ppw  16190  ppiqub  16194  2lgsoddprmlem3c  16326  2lgsoddprmlem3d  16327  nninfomni  17160