Theorem tpeq1d 4123

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 4120	. 2 |- ( A = B -> { A , C , D } = { B , C , D } )
3	1, 2	syl 16	1 |- ( ph -> { A , C , D } = { B , C , D } )

Syntax hints:   -> wi 4   = wceq 1395   {ctp 4036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-un 3476  df-sn 4033  df-pr 4035  df-tp 4037

This theorem is referenced by:  tpeq123d  4126