Theorem oteq2 3877

Description: Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)

Assertion

Ref	Expression
oteq2	|- ( A = B -> <. C , A , D >. = <. C , B , D >. )

Proof of Theorem oteq2

Step	Hyp	Ref	Expression
1		opeq2 3868	. . 3 |- ( A = B -> <. C , A >. = <. C , B >. )
2	1	opeq1d 3873	. 2 |- ( A = B -> <. <. C , A >. , D >. = <. <. C , B >. , D >. )
3		df-ot 3683	. 2 |- <. C , A , D >. = <. <. C , A >. , D >.
4		df-ot 3683	. 2 |- <. C , B , D >. = <. <. C , B >. , D >.
5	2, 3, 4	3eqtr4g 2289	1 |- ( A = B -> <. C , A , D >. = <. C , B , D >. )

Syntax hints:   -> wi 4   = wceq 1398   <.cop 3676   <.cotp 3677

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-ot 3683

This theorem is referenced by:  oteq2d  3880