Theorem preq1 2448

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 2417	. . 3 |- (A = B -> {A} = {B})
2	1	uneq1d 2183	. 2 |- (A = B -> ({A} u. {C}) = ({B} u. {C}))
3		df-pr 2413	. 2 |- {A, C} = ({A} u. {C})
4		df-pr 2413	. 2 |- {B, C} = ({B} u. {C})
5	2, 3, 4	3eqtr4g 1531	1 |- (A = B -> {A, C} = {B, C})

Syntax hints:   -> wi 3   = wceq 956   u. cun 2045  {csn 2409  {cpr 2410

This theorem is referenced by:  preq2 2449  prssg 2472  preq12b 2483  opeq1 2487  opprc1 2498  uniprg 2516  prex 2781  opthwiener 2807  relop 3275  funopg 3547  opthreg 4604  aceq6b 4742  brdom7disj 4804  brdom6disj 4805  metxpdval 7829  sshjval3t 9326  intprd 10471

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-un 2050  df-sn 2412  df-pr 2413

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