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Theorem preq1 3438
Description: Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998.)
Assertion
Ref Expression
preq1 (A = B → {A, 𝐶} = {B, 𝐶})

Proof of Theorem preq1
StepHypRef Expression
1 sneq 3378 . . 3 (A = B → {A} = {B})
21uneq1d 3090 . 2 (A = B → ({A} ∪ {𝐶}) = ({B} ∪ {𝐶}))
3 df-pr 3374 . 2 {A, 𝐶} = ({A} ∪ {𝐶})
4 df-pr 3374 . 2 {B, 𝐶} = ({B} ∪ {𝐶})
52, 3, 43eqtr4g 2094 1 (A = B → {A, 𝐶} = {B, 𝐶})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  cun 2909  {csn 3367  {cpr 3368
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374
This theorem is referenced by:  preq2  3439  preq12  3440  preq1i  3441  preq1d  3444  tpeq1  3447  prnzg  3483  preq12b  3532  preq12bg  3535  opeq1  3540  uniprg  3586  intprg  3639  prexgOLD  3937  prexg  3938  opthreg  4234  bj-prexg  9366
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