Theorem tpeq123d 3735

Description: Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)

Hypotheses

Ref	Expression
tpeq1d.1	|- ( ph -> A = B )
tpeq123d.2	|- ( ph -> C = D )
tpeq123d.3	|- ( ph -> E = F )

Assertion

Ref	Expression
tpeq123d	|- ( ph -> { A , C , E } = { B , D , F } )

Proof of Theorem tpeq123d

Step	Hyp	Ref	Expression
1		tpeq1d.1	. . 3 |- ( ph -> A = B )
2	1	tpeq1d 3732	. 2 |- ( ph -> { A , C , E } = { B , C , E } )
3		tpeq123d.2	. . 3 |- ( ph -> C = D )
4	3	tpeq2d 3733	. 2 |- ( ph -> { B , C , E } = { B , D , E } )
5		tpeq123d.3	. . 3 |- ( ph -> E = F )
6	5	tpeq3d 3734	. 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   {ctp 3645

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-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-tp 3651

This theorem is referenced by:  fz0tp  10279  fz0to4untppr  10281  fzo0to3tp  10385  prdsex  13216  prdsval  13220  imasex  13252  imasival  13253  psrval  14543