Theorem 3eqtr2i 2223

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 2220	. 2 |- A = C
4		3eqtr2i.3	. 2 |- C = D
5	3, 4	eqtri 2217	1 |- A = D

Syntax hints:   = wceq 1364

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 1461  ax-gen 1463  ax-4 1524  ax-17 1540  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189

This theorem is referenced by:  dfrab3  3439  iunid  3972  cnvcnv  5122  cocnvcnv2  5181  fmptap  5752  exmidfodomrlemim  7268  negdii  8310  halfpm6th  9211  numma  9500  numaddc  9504  6p5lem  9526  8p2e10  9536  binom2i  10740  0.999...  11686  flodddiv4  12101  6gcd4e2  12162  dfphi2  12388  karatsuba  12599  cosq23lt0  15069  pigt3  15080  1sgm2ppw  15231  2lgsoddprmlem3c  15350  2lgsoddprmlem3d  15351  nninfomni  15663