Theorem oteq123d 3834

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 3831	. 2 |- ( ph -> <. A , C , E >. = <. B , C , E >. )
3		oteq123d.2	. . 3 |- ( ph -> C = D )
4	3	oteq2d 3832	. 2 |- ( ph -> <. B , C , E >. = <. B , D , E >. )
5		oteq123d.3	. . 3 |- ( ph -> E = F )
6	5	oteq3d 3833	. 2 |- ( ph -> <. B , D , E >. = <. B , D , F >. )
7	2, 4, 6	3eqtrd 2242	1 |- ( ph -> <. A , C , E >. = <. B , D , F >. )

Syntax hints:   -> wi 4   = wceq 1373   <.cotp 3637

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-ot 3643

This theorem is referenced by: (None)