Theorem tpeq2 4682

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 4673	. . 3 ⊢ (𝐴 = 𝐵 → {𝐶, 𝐴} = {𝐶, 𝐵})
2	1	uneq1d 4141	. 2 ⊢ (𝐴 = 𝐵 → ({𝐶, 𝐴} ∪ {𝐷}) = ({𝐶, 𝐵} ∪ {𝐷}))
3		df-tp 4575	. 2 ⊢ {𝐶, 𝐴, 𝐷} = ({𝐶, 𝐴} ∪ {𝐷})
4		df-tp 4575	. 2 ⊢ {𝐶, 𝐵, 𝐷} = ({𝐶, 𝐵} ∪ {𝐷})
5	2, 3, 4	3eqtr4g 2884	1 ⊢ (𝐴 = 𝐵 → {𝐶, 𝐴, 𝐷} = {𝐶, 𝐵, 𝐷})

Syntax hints:   → wi 4   = wceq 1536   ∪ cun 3937  {csn 4570  {cpr 4572  {ctp 4574

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-v 3499  df-un 3944  df-sn 4571  df-pr 4573  df-tp 4575

This theorem is referenced by:  tpeq2d  4685  fntpb  6975  fztpval  12972  hashtpg  13846  dvh4dimN  38587  lmod1  44554