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Theorem tpeq123d 3673
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 3670 . 2 (𝜑 → {𝐴, 𝐶, 𝐸} = {𝐵, 𝐶, 𝐸})
3 tpeq123d.2 . . 3 (𝜑𝐶 = 𝐷)
43tpeq2d 3671 . 2 (𝜑 → {𝐵, 𝐶, 𝐸} = {𝐵, 𝐷, 𝐸})
5 tpeq123d.3 . . 3 (𝜑𝐸 = 𝐹)
65tpeq3d 3672 . 2 (𝜑 → {𝐵, 𝐷, 𝐸} = {𝐵, 𝐷, 𝐹})
72, 4, 63eqtrd 2207 1 (𝜑 → {𝐴, 𝐶, 𝐸} = {𝐵, 𝐷, 𝐹})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1348  {ctp 3583
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3587  df-pr 3588  df-tp 3589
This theorem is referenced by:  fz0tp  10071  fz0to4untppr  10073  fzo0to3tp  10168
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