Theorem tpeq123d 3724

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 3721	. 2 |- ( ph -> { A , C , E } = { B , C , E } )
3		tpeq123d.2	. . 3 |- ( ph -> C = D )
4	3	tpeq2d 3722	. 2 |- ( ph -> { B , C , E } = { B , D , E } )
5		tpeq123d.3	. . 3 |- ( ph -> E = F )
6	5	tpeq3d 3723	. 2 |- ( ph -> { B , D , E } = { B , D , F } )
7	2, 4, 6	3eqtrd 2241	1 |- ( ph -> { A , C , E } = { B , D , F } )

Syntax hints:   -> wi 4   = wceq 1372   {ctp 3634

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169  df-sn 3638  df-pr 3639  df-tp 3640

This theorem is referenced by:  fz0tp  10243  fz0to4untppr  10245  fzo0to3tp  10346  prdsex  13072  prdsval  13076  imasex  13108  imasival  13109  psrval  14399