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Theorem tpeq123d 3814
 Description: Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
Hypotheses
Ref Expression
tpeq1d.1 (φA = B)
tpeq123d.2 (φC = D)
tpeq123d.3 (φE = F)
Assertion
Ref Expression
tpeq123d (φ → {A, C, E} = {B, D, F})

Proof of Theorem tpeq123d
StepHypRef Expression
1 tpeq1d.1 . . 3 (φA = B)
21tpeq1d 3811 . 2 (φ → {A, C, E} = {B, C, E})
3 tpeq123d.2 . . 3 (φC = D)
43tpeq2d 3812 . 2 (φ → {B, C, E} = {B, D, E})
5 tpeq123d.3 . . 3 (φE = F)
65tpeq3d 3813 . 2 (φ → {B, D, E} = {B, D, F})
72, 4, 63eqtrd 2389 1 (φ → {A, C, E} = {B, D, F})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642  {ctp 3739 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742  df-tp 3743 This theorem is referenced by: (None)
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