Theorem oteq123d 3823

Description: Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

Ref	Expression
oteq1d.1	|- ( ph -> A = B )
oteq123d.2	|- ( ph -> C = D )
oteq123d.3	|- ( ph -> E = F )

Assertion

Ref	Expression
oteq123d	|- ( ph -> <. A , C , E >. = <. B , D , F >. )

Proof of Theorem oteq123d

Step	Hyp	Ref	Expression
1		oteq1d.1	. . 3 |- ( ph -> A = B )
2	1	oteq1d 3820	. 2 |- ( ph -> <. A , C , E >. = <. B , C , E >. )
3		oteq123d.2	. . 3 |- ( ph -> C = D )
4	3	oteq2d 3821	. 2 |- ( ph -> <. B , C , E >. = <. B , D , E >. )
5		oteq123d.3	. . 3 |- ( ph -> E = F )
6	5	oteq3d 3822	. 2 |- ( ph -> <. B , D , E >. = <. B , D , F >. )
7	2, 4, 6	3eqtrd 2233	1 |- ( ph -> <. A , C , E >. = <. B , D , F >. )

Syntax hints:   -> wi 4   = wceq 1364   <.cotp 3626

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631  df-ot 3632

This theorem is referenced by: (None)