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Theorem preq2b 4852
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 4737 . . 3 {𝐶, 𝐴} = {𝐴, 𝐶}
2 prcom 4737 . . 3 {𝐶, 𝐵} = {𝐵, 𝐶}
31, 2eqeq12i 2753 . 2 ({𝐶, 𝐴} = {𝐶, 𝐵} ↔ {𝐴, 𝐶} = {𝐵, 𝐶})
4 preq1b.a . . 3 (𝜑𝐴𝑉)
5 preq1b.b . . 3 (𝜑𝐵𝑊)
64, 5preq1b 4851 . 2 (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵))
73, 6bitrid 283 1 (𝜑 → ({𝐶, 𝐴} = {𝐶, 𝐵} ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wcel 2106  {cpr 4633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-sn 4632  df-pr 4634
This theorem is referenced by:  umgr2v2enb1  29559  clsk1indlem4  44034  usgrexmpl2nb0  47926  usgrexmpl2nb1  47927  usgrexmpl2nb2  47928  usgrexmpl2nb3  47929  usgrexmpl2nb4  47930  usgrexmpl2nb5  47931
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