Theorem tpeq3 4674

Description: Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)

Assertion

Ref	Expression
tpeq3	⊢ (𝐴 = 𝐵 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵})

Proof of Theorem tpeq3

Step	Hyp	Ref	Expression
1		sneq 4571	. . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵})
2	1	uneq2d 4139	. 2 ⊢ (𝐴 = 𝐵 → ({𝐶, 𝐷} ∪ {𝐴}) = ({𝐶, 𝐷} ∪ {𝐵}))
3		df-tp 4566	. 2 ⊢ {𝐶, 𝐷, 𝐴} = ({𝐶, 𝐷} ∪ {𝐴})
4		df-tp 4566	. 2 ⊢ {𝐶, 𝐷, 𝐵} = ({𝐶, 𝐷} ∪ {𝐵})
5	2, 3, 4	3eqtr4g 2881	1 ⊢ (𝐴 = 𝐵 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵})

Syntax hints:   → wi 4   = wceq 1533   ∪ cun 3934  {csn 4561  {cpr 4563  {ctp 4565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3497  df-un 3941  df-sn 4562  df-tp 4566

This theorem is referenced by:  tpeq3d  4677  tppreq3  4689  fntpb  6966  fztpval  12963  hashtpg  13837  dvh4dimN  38577