Theorem 3eqtr2i 2258

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 2255	. 2 ⊢ 𝐴 = 𝐶
4		3eqtr2i.3	. 2 ⊢ 𝐶 = 𝐷
5	3, 4	eqtri 2252	1 ⊢ 𝐴 = 𝐷

Syntax hints:   = wceq 1398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-gen 1498  ax-4 1559  ax-17 1575  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-cleq 2224

This theorem is referenced by:  dfrab3  3485  iunid  4031  cnvcnv  5196  cocnvcnv2  5255  fmptap  5852  exmidfodomrlemim  7455  negdii  8505  halfpm6th  9406  numma  9698  numaddc  9702  6p5lem  9724  8p2e10  9734  binom2i  10956  0.999...  12145  flodddiv4  12560  6gcd4e2  12629  dfphi2  12855  karatsuba  13066  cosq23lt0  15627  pigt3  15638  1sgm2ppw  15792  2lgsoddprmlem3c  15911  2lgsoddprmlem3d  15912  nninfomni  16728