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Theorem tpeq123d 3615
 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 3612 . 2 (𝜑 → {𝐴, 𝐶, 𝐸} = {𝐵, 𝐶, 𝐸})
3 tpeq123d.2 . . 3 (𝜑𝐶 = 𝐷)
43tpeq2d 3613 . 2 (𝜑 → {𝐵, 𝐶, 𝐸} = {𝐵, 𝐷, 𝐸})
5 tpeq123d.3 . . 3 (𝜑𝐸 = 𝐹)
65tpeq3d 3614 . 2 (𝜑 → {𝐵, 𝐷, 𝐸} = {𝐵, 𝐷, 𝐹})
72, 4, 63eqtrd 2176 1 (𝜑 → {𝐴, 𝐶, 𝐸} = {𝐵, 𝐷, 𝐹})
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1331  {ctp 3529 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-tp 3535 This theorem is referenced by:  fz0tp  9913  fzo0to3tp  10008
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