Theorem oteq2 3909

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 3900	. . 3 |- ( A = B -> <. C , A >. = <. C , B >. )
2	1	opeq1d 3905	. 2 |- ( A = B -> <. <. C , A >. , D >. = <. <. C , B >. , D >. )
3		df-ot 3715	. 2 |- <. C , A , D >. = <. <. C , A >. , D >.
4		df-ot 3715	. 2 |- <. C , B , D >. = <. <. C , B >. , D >.
5	2, 3, 4	3eqtr4g 2296	1 |- ( A = B -> <. C , A , D >. = <. C , B , D >. )

Syntax hints:   -> wi 4   = wceq 1402   <.cop 3708   <.cotp 3709

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3711  df-pr 3712  df-op 3714  df-ot 3715

This theorem is referenced by:  oteq2d  3912