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Theorem preq2b 4847
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 4732 . . 3 {𝐶, 𝐴} = {𝐴, 𝐶}
2 prcom 4732 . . 3 {𝐶, 𝐵} = {𝐵, 𝐶}
31, 2eqeq12i 2755 . 2 ({𝐶, 𝐴} = {𝐶, 𝐵} ↔ {𝐴, 𝐶} = {𝐵, 𝐶})
4 preq1b.a . . 3 (𝜑𝐴𝑉)
5 preq1b.b . . 3 (𝜑𝐵𝑊)
64, 5preq1b 4846 . 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 1540  wcel 2108  {cpr 4628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-un 3956  df-sn 4627  df-pr 4629
This theorem is referenced by:  umgr2v2enb1  29544  clsk1indlem4  44057  usgrexmpl2nb0  47990  usgrexmpl2nb1  47991  usgrexmpl2nb2  47992  usgrexmpl2nb3  47993  usgrexmpl2nb4  47994  usgrexmpl2nb5  47995
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