Theorem tpeq3 4424

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 4332	. . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵})
2	1	uneq2d 3911	. 2 ⊢ (𝐴 = 𝐵 → ({𝐶, 𝐷} ∪ {𝐴}) = ({𝐶, 𝐷} ∪ {𝐵}))
3		df-tp 4327	. 2 ⊢ {𝐶, 𝐷, 𝐴} = ({𝐶, 𝐷} ∪ {𝐴})
4		df-tp 4327	. 2 ⊢ {𝐶, 𝐷, 𝐵} = ({𝐶, 𝐷} ∪ {𝐵})
5	2, 3, 4	3eqtr4g 2820	1 ⊢ (𝐴 = 𝐵 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵})

Syntax hints:   → wi 4   = wceq 1632   ∪ cun 3714  {csn 4322  {cpr 4324  {ctp 4326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-v 3343  df-un 3721  df-sn 4323  df-tp 4327

This theorem is referenced by:  tpeq3d  4427  tppreq3  4439  fntpb  6639  fztpval  12616  hashtpg  13480  dvh4dimN  37257