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Theorem preq2b 4519
Description: Biconditional equality lemma for unordered pairs, deduction form. Two unordered pairs have the same first element iff the second elements are equal. (Contributed by AV, 18-Dec-2020.)
Hypotheses
Ref Expression
preq1b.a (𝜑𝐴𝑉)
preq1b.b (𝜑𝐵𝑊)
Assertion
Ref Expression
preq2b (𝜑 → ({𝐶, 𝐴} = {𝐶, 𝐵} ↔ 𝐴 = 𝐵))

Proof of Theorem preq2b
StepHypRef Expression
1 prcom 4407 . . 3 {𝐶, 𝐴} = {𝐴, 𝐶}
2 prcom 4407 . . 3 {𝐶, 𝐵} = {𝐵, 𝐶}
31, 2eqeq12i 2770 . 2 ({𝐶, 𝐴} = {𝐶, 𝐵} ↔ {𝐴, 𝐶} = {𝐵, 𝐶})
4 preq1b.a . . 3 (𝜑𝐴𝑉)
5 preq1b.b . . 3 (𝜑𝐵𝑊)
64, 5preq1b 4518 . 2 (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵))
73, 6syl5bb 272 1 (𝜑 → ({𝐶, 𝐴} = {𝐶, 𝐵} ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1628  wcel 2135  {cpr 4319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-v 3338  df-un 3716  df-sn 4318  df-pr 4320
This theorem is referenced by:  umgr2v2enb1  26628  clsk1indlem4  38840
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