Theorem 3eqtr2i 2256

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 2253	. 2 ⊢ 𝐴 = 𝐶
4		3eqtr2i.3	. 2 ⊢ 𝐶 = 𝐷
5	3, 4	eqtri 2250	1 ⊢ 𝐴 = 𝐷

Syntax hints:   = wceq 1395

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 1493  ax-gen 1495  ax-4 1556  ax-17 1572  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222

This theorem is referenced by:  dfrab3  3480  iunid  4020  cnvcnv  5180  cocnvcnv2  5239  fmptap  5828  exmidfodomrlemim  7375  negdii  8426  halfpm6th  9327  numma  9617  numaddc  9621  6p5lem  9643  8p2e10  9653  binom2i  10865  0.999...  12027  flodddiv4  12442  6gcd4e2  12511  dfphi2  12737  karatsuba  12948  cosq23lt0  15501  pigt3  15512  1sgm2ppw  15663  2lgsoddprmlem3c  15782  2lgsoddprmlem3d  15783  nninfomni  16344