Theorem tpeq2 4695

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

Assertion

Ref	Expression
tpeq2	⊢ (𝐴 = 𝐵 → {𝐶, 𝐴, 𝐷} = {𝐶, 𝐵, 𝐷})

Proof of Theorem tpeq2

Step	Hyp	Ref	Expression
1		preq2 4686	. . 3 ⊢ (𝐴 = 𝐵 → {𝐶, 𝐴} = {𝐶, 𝐵})
2	1	uneq1d 4118	. 2 ⊢ (𝐴 = 𝐵 → ({𝐶, 𝐴} ∪ {𝐷}) = ({𝐶, 𝐵} ∪ {𝐷}))
3		df-tp 4582	. 2 ⊢ {𝐶, 𝐴, 𝐷} = ({𝐶, 𝐴} ∪ {𝐷})
4		df-tp 4582	. 2 ⊢ {𝐶, 𝐵, 𝐷} = ({𝐶, 𝐵} ∪ {𝐷})
5	2, 3, 4	3eqtr4g 2789	1 ⊢ (𝐴 = 𝐵 → {𝐶, 𝐴, 𝐷} = {𝐶, 𝐵, 𝐷})

Syntax hints:   → wi 4   = wceq 1540   ∪ cun 3901  {csn 4577  {cpr 4579  {ctp 4581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3438  df-un 3908  df-sn 4578  df-pr 4580  df-tp 4582

This theorem is referenced by:  tpeq2d  4698  fntpb  7145  fztpval  13489  hashtpg  14392  hash3tpde  14400  dvh4dimN  41426  cycl3grtri  47931  grimgrtri  47933  usgrgrtrirex  47934  grlimgrtri  47987  usgrexmpl1tri  48009  lmod1  48477