Theorem oteq2 4412

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 4403	. . 3 |- ( A = B -> <. C , A >. = <. C , B >. )
2	1	opeq1d 4408	. 2 |- ( A = B -> <. <. C , A >. , D >. = <. <. C , B >. , D >. )
3		df-ot 4186	. 2 |- <. C , A , D >. = <. <. C , A >. , D >.
4		df-ot 4186	. 2 |- <. C , B , D >. = <. <. C , B >. , D >.
5	2, 3, 4	3eqtr4g 2681	1 |- ( A = B -> <. C , A , D >. = <. C , B , D >. )

Syntax hints:   -> wi 4   = wceq 1483   <.cop 4183   <.cotp 4185

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-ot 4186

This theorem is referenced by:  oteq2d  4415  efgi  18132  efgtf  18135  efgtval  18136