ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  preq2d GIF version

Theorem preq2d 3448
Description: Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
Hypothesis
Ref Expression
preq1d.1 (φA = B)
Assertion
Ref Expression
preq2d (φ → {𝐶, A} = {𝐶, B})

Proof of Theorem preq2d
StepHypRef Expression
1 preq1d.1 . 2 (φA = B)
2 preq2 3442 . 2 (A = B → {𝐶, A} = {𝐶, B})
31, 2syl 14 1 (φ → {𝐶, A} = {𝐶, B})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1243  {cpr 3371
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2556  df-un 2919  df-sn 3376  df-pr 3377
This theorem is referenced by:  opthreg  4237
  Copyright terms: Public domain W3C validator