Theorem tpeq123d 3710

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 3707	. 2 |- ( ph -> { A , C , E } = { B , C , E } )
3		tpeq123d.2	. . 3 |- ( ph -> C = D )
4	3	tpeq2d 3708	. 2 |- ( ph -> { B , C , E } = { B , D , E } )
5		tpeq123d.3	. . 3 |- ( ph -> E = F )
6	5	tpeq3d 3709	. 2 |- ( ph -> { B , D , E } = { B , D , F } )
7	2, 4, 6	3eqtrd 2230	1 |- ( ph -> { A , C , E } = { B , D , F } )

Syntax hints:   -> wi 4   = wceq 1364   {ctp 3620

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-tp 3626

This theorem is referenced by:  fz0tp  10188  fz0to4untppr  10190  fzo0to3tp  10286  prdsex  12880  imasex  12888  imasival  12889  psrval  14152