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Theorem preqsnOLD 4529
Description: Obsolete proof of preqsn 4528 as of 12-Jun-2022. (Contributed by NM, 3-Jun-2008.) (Proof shortened by JJ, 23-Jul-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
preqsn.1 𝐴 ∈ V
preqsn.2 𝐵 ∈ V
preqsnOLD.3 𝐶 ∈ V
Assertion
Ref Expression
preqsnOLD ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵𝐵 = 𝐶))

Proof of Theorem preqsnOLD
StepHypRef Expression
1 dfsn2 4330 . . 3 {𝐶} = {𝐶, 𝐶}
21eqeq2i 2783 . 2 ({𝐴, 𝐵} = {𝐶} ↔ {𝐴, 𝐵} = {𝐶, 𝐶})
3 oridm 882 . . 3 (((𝐴 = 𝐶𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐶)) ↔ (𝐴 = 𝐶𝐵 = 𝐶))
4 preqsn.1 . . . 4 𝐴 ∈ V
5 preqsn.2 . . . 4 𝐵 ∈ V
6 preqsnOLD.3 . . . 4 𝐶 ∈ V
74, 5, 6, 6preq12b 4514 . . 3 ({𝐴, 𝐵} = {𝐶, 𝐶} ↔ ((𝐴 = 𝐶𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐶)))
8 eqeq2 2782 . . . 4 (𝐵 = 𝐶 → (𝐴 = 𝐵𝐴 = 𝐶))
98pm5.32ri 559 . . 3 ((𝐴 = 𝐵𝐵 = 𝐶) ↔ (𝐴 = 𝐶𝐵 = 𝐶))
103, 7, 93bitr4i 292 . 2 ({𝐴, 𝐵} = {𝐶, 𝐶} ↔ (𝐴 = 𝐵𝐵 = 𝐶))
112, 10bitri 264 1 ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵𝐵 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382  wo 828   = wceq 1631  wcel 2145  Vcvv 3351  {csn 4317  {cpr 4319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-un 3729  df-sn 4318  df-pr 4320
This theorem is referenced by: (None)
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