Theorem 3eqtr2i 2261

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 2258	. 2 ⊢ 𝐴 = 𝐶
4		3eqtr2i.3	. 2 ⊢ 𝐶 = 𝐷
5	3, 4	eqtri 2255	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 2216

This theorem depends on definitions:  df-bi 117  df-cleq 2227

This theorem is referenced by:  dfrab3  3501  iunid  4052  cnvcnv  5220  cocnvcnv2  5279  fmptap  5879  exmidfodomrlemim  7517  negdii  8573  halfpm6th  9475  numma  9770  numaddc  9774  6p5lem  9796  8p2e10  9806  binom2i  11034  0.999...  12232  flodddiv4  12647  6gcd4e2  12716  dfphi2  12942  karatsuba  13153  ballotfilem1  13164  ballotfilemfval0  13179  ballotfilemth  13225  cosq23lt0  15824  pigt3  15835  1sgm2ppw  15989  2lgsoddprmlem3c  16108  2lgsoddprmlem3d  16109  nninfomni  16923