Theorem preq1 2452

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 2421	. . 3 |- (A = B -> {A} = {B})
2	1	uneq1d 2186	. 2 |- (A = B -> ({A} u. {C}) = ({B} u. {C}))
3		df-pr 2417	. 2 |- {A, C} = ({A} u. {C})
4		df-pr 2417	. 2 |- {B, C} = ({B} u. {C})
5	2, 3, 4	3eqtr4g 1534	1 |- (A = B -> {A, C} = {B, C})

Syntax hints:   -> wi 3   = wceq 958   u. cun 2048  {csn 2413  {cpr 2414

This theorem is referenced by:  preq2 2453  prssg 2476  preq12b 2487  opeq1 2491  opprc1 2502  uniprg 2520  prex 2787  opthwiener 2813  relop 3281  funopg 3553  opthreg 4613  aceq6b 4752  brdom7disj 4814  brdom6disj 4815  metxpdval 7826  sshjval3t 9321  intprd 10461

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-un 2053  df-sn 2416  df-pr 2417

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