Theorem oteq123d 3564

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 3561	. 2 |- ( ph -> <. A , C , E >. = <. B , C , E >. )
3		oteq123d.2	. . 3 |- ( ph -> C = D )
4	3	oteq2d 3562	. 2 |- ( ph -> <. B , C , E >. = <. B , D , E >. )
5		oteq123d.3	. . 3 |- ( ph -> E = F )
6	5	oteq3d 3563	. 2 |- ( ph -> <. B , D , E >. = <. B , D , F >. )
7	2, 4, 6	3eqtrd 2076	1 |- ( ph -> <. A , C , E >. = <. B , D , F >. )

Syntax hints:   -> wi 4   = wceq 1243   <.cotp 3379

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-ot 3385

This theorem is referenced by: (None)