Theorem tpeq1d 3696

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

Hypothesis

Ref	Expression
tpeq1d.1	|- ( ph -> A = B )

Assertion

Ref	Expression
tpeq1d	|- ( ph -> { A , C , D } = { B , C , D } )

Proof of Theorem tpeq1d

Step	Hyp	Ref	Expression
1		tpeq1d.1	. 2 |- ( ph -> A = B )
2		tpeq1 3693	. 2 |- ( A = B -> { A , C , D } = { B , C , D } )
3	1, 2	syl 14	1 |- ( ph -> { A , C , D } = { B , C , D } )

Syntax hints:   -> wi 4   = wceq 1364   {ctp 3609

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 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-tp 3615

This theorem is referenced by:  tpeq123d  3699