Theorem oteq123d 3781

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 3778	. 2 |- ( ph -> <. A , C , E >. = <. B , C , E >. )
3		oteq123d.2	. . 3 |- ( ph -> C = D )
4	3	oteq2d 3779	. 2 |- ( ph -> <. B , C , E >. = <. B , D , E >. )
5		oteq123d.3	. . 3 |- ( ph -> E = F )
6	5	oteq3d 3780	. 2 |- ( ph -> <. B , D , E >. = <. B , D , F >. )
7	2, 4, 6	3eqtrd 2208	1 |- ( ph -> <. A , C , E >. = <. B , D , F >. )

Syntax hints:   -> wi 4   = wceq 1349   <.cotp 3588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-ext 2153

This theorem depends on definitions:  df-bi 116  df-3an 976  df-tru 1352  df-nf 1455  df-sb 1757  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-v 2733  df-un 3126  df-sn 3590  df-pr 3591  df-op 3593  df-ot 3594

This theorem is referenced by: (None)