Theorem tppreq3 3695

Description: An unordered triple is an unordered pair if one of its elements is identical with another element. (Contributed by Alexander van der Vekens, 6-Oct-2017.)

Assertion

Ref	Expression
tppreq3	⊢ (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})

Proof of Theorem tppreq3

Step	Hyp	Ref	Expression
1		tpeq3 3680	. . 3 ⊢ (𝐶 = 𝐵 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵, 𝐵})
2	1	eqcoms 2180	. 2 ⊢ (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵, 𝐵})
3		tpidm23 3693	. 2 ⊢ {𝐴, 𝐵, 𝐵} = {𝐴, 𝐵}
4	2, 3	eqtrdi 2226	1 ⊢ (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})

Syntax hints:   → wi 4   = wceq 1353  {cpr 3593  {ctp 3594

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3or 979  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599  df-tp 3600

This theorem is referenced by: (None)