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  5243  cocnvcnv2  5302  fmptap  5908  exmidfodomrlemim  7559  negdii  8617  halfpm6th  9536  numma  9835  numaddc  9839  6p5lem  9861  8p2e10  9871  binom2i  11105  0.999...  12312  flodddiv4  12727  6gcd4e2  12796  dfphi2  13026  mod2xnegi  13226  karatsuba  13238  1259lem1  13270  ballotfilem1  13277  ballotfilemfval0  13292  ballotfilemth  13338  cosq23lt0  16077  pigt3  16088  log2ublem3  16240  1sgm2ppw  16306  ppiqub  16310  bposlem8  16335  bposlem9  16336  2lgsoddprmlem3c  16450  2lgsoddprmlem3d  16451  nninfomni  17285