Theorem oteq2 3872

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 3863	. . 3 |- ( A = B -> <. C , A >. = <. C , B >. )
2	1	opeq1d 3868	. 2 |- ( A = B -> <. <. C , A >. , D >. = <. <. C , B >. , D >. )
3		df-ot 3679	. 2 |- <. C , A , D >. = <. <. C , A >. , D >.
4		df-ot 3679	. 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 1397   <.cop 3672   <.cotp 3673

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-ot 3679

This theorem is referenced by:  oteq2d  3875