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Theorem preqsnOLD 4364
Description: Obsolete proof of preqsn 4363 as of 23-Jul-2021. (Contributed by NM, 3-Jun-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
preqsn.1 𝐴 ∈ V
preqsn.2 𝐵 ∈ V
preqsn.3 𝐶 ∈ V
Assertion
Ref Expression
preqsnOLD ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵𝐵 = 𝐶))

Proof of Theorem preqsnOLD
StepHypRef Expression
1 dfsn2 4163 . . 3 {𝐶} = {𝐶, 𝐶}
21eqeq2i 2633 . 2 ({𝐴, 𝐵} = {𝐶} ↔ {𝐴, 𝐵} = {𝐶, 𝐶})
3 preqsn.1 . . . 4 𝐴 ∈ V
4 preqsn.2 . . . 4 𝐵 ∈ V
5 preqsn.3 . . . 4 𝐶 ∈ V
63, 4, 5, 5preq12b 4352 . . 3 ({𝐴, 𝐵} = {𝐶, 𝐶} ↔ ((𝐴 = 𝐶𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐶)))
7 oridm 536 . . . 4 (((𝐴 = 𝐶𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐶)) ↔ (𝐴 = 𝐶𝐵 = 𝐶))
8 eqtr3 2642 . . . . . 6 ((𝐴 = 𝐶𝐵 = 𝐶) → 𝐴 = 𝐵)
9 simpr 477 . . . . . 6 ((𝐴 = 𝐶𝐵 = 𝐶) → 𝐵 = 𝐶)
108, 9jca 554 . . . . 5 ((𝐴 = 𝐶𝐵 = 𝐶) → (𝐴 = 𝐵𝐵 = 𝐶))
11 eqtr 2640 . . . . . 6 ((𝐴 = 𝐵𝐵 = 𝐶) → 𝐴 = 𝐶)
12 simpr 477 . . . . . 6 ((𝐴 = 𝐵𝐵 = 𝐶) → 𝐵 = 𝐶)
1311, 12jca 554 . . . . 5 ((𝐴 = 𝐵𝐵 = 𝐶) → (𝐴 = 𝐶𝐵 = 𝐶))
1410, 13impbii 199 . . . 4 ((𝐴 = 𝐶𝐵 = 𝐶) ↔ (𝐴 = 𝐵𝐵 = 𝐶))
157, 14bitri 264 . . 3 (((𝐴 = 𝐶𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐶)) ↔ (𝐴 = 𝐵𝐵 = 𝐶))
166, 15bitri 264 . 2 ({𝐴, 𝐵} = {𝐶, 𝐶} ↔ (𝐴 = 𝐵𝐵 = 𝐶))
172, 16bitri 264 1 ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵𝐵 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 383  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  {csn 4150  {cpr 4152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-un 3561  df-sn 4151  df-pr 4153
This theorem is referenced by: (None)
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