Theorem oteq2 3803

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 3794	. . 3 |- ( A = B -> <. C , A >. = <. C , B >. )
2	1	opeq1d 3799	. 2 |- ( A = B -> <. <. C , A >. , D >. = <. <. C , B >. , D >. )
3		df-ot 3617	. 2 |- <. C , A , D >. = <. <. C , A >. , D >.
4		df-ot 3617	. 2 |- <. C , B , D >. = <. <. C , B >. , D >.
5	2, 3, 4	3eqtr4g 2247	1 |- ( A = B -> <. C , A , D >. = <. C , B , D >. )

Syntax hints:   -> wi 4   = wceq 1364   <.cop 3610   <.cotp 3611

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-ot 3617

This theorem is referenced by:  oteq2d  3806