Theorem preq1 1870

Description: An equality theorem for unordered pairs.

Assertion

Ref	Expression
preq1	⊢ (A = B → {A, C} = {B, C})

Proof of Theorem preq1

Step	Hyp	Ref	Expression
1		sneq 1816	. . 3 ⊢ (A = B → {A} = {B})
2	1	uneq1d 1610	. 2 ⊢ (A = B → ({A} ∪ {C}) = ({B} ∪ {C}))
3		df-pr 1812	. 2 ⊢ {A, C} = ({A} ∪ {C})
4		df-pr 1812	. 2 ⊢ {B, C} = ({B} ∪ {C})
5	2, 3, 4	3eqtr4g 1147	1 ⊢ (A = B → {A, C} = {B, C})

Syntax hints:   → wi 2   = wceq 1091   ∪ cun 1485  {csn 1808  {cpr 1809

This theorem is referenced by:  preq2 1871  preq12b 1874  opeq1 1876  prex 1892  opprc1 1905  opthwiener 1914  opthreg 3455  aceq6b 3565  sshjval3t 5327

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812

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