Theorem tpeq3d 4744

Description: Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)

Hypothesis

Ref	Expression
tpeq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
tpeq3d	⊢ (𝜑 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵})

Proof of Theorem tpeq3d

Step	Hyp	Ref	Expression
1		tpeq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		tpeq3 4741	. 2 ⊢ (𝐴 = 𝐵 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵})
3	1, 2	syl 17	1 ⊢ (𝜑 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵})

Syntax hints:   → wi 4   = wceq 1533  {ctp 4625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-un 3946  df-sn 4622  df-tp 4626

This theorem is referenced by:  tpeq123d  4745  fntpb  7203  erngset  40165  erngset-rN  40173