Theorem oteq123d 3848

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 3845	. 2 |- ( ph -> <. A , C , E >. = <. B , C , E >. )
3		oteq123d.2	. . 3 |- ( ph -> C = D )
4	3	oteq2d 3846	. 2 |- ( ph -> <. B , C , E >. = <. B , D , E >. )
5		oteq123d.3	. . 3 |- ( ph -> E = F )
6	5	oteq3d 3847	. 2 |- ( ph -> <. B , D , E >. = <. B , D , F >. )
7	2, 4, 6	3eqtrd 2244	1 |- ( ph -> <. A , C , E >. = <. B , D , F >. )

Syntax hints:   -> wi 4   = wceq 1373   <.cotp 3647

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652  df-ot 3653

This theorem is referenced by: (None)