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Theorem preq2b 4318
 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 4211 . . 3 {𝐶, 𝐴} = {𝐴, 𝐶}
2 prcom 4211 . . 3 {𝐶, 𝐵} = {𝐵, 𝐶}
31, 2eqeq12i 2624 . 2 ({𝐶, 𝐴} = {𝐶, 𝐵} ↔ {𝐴, 𝐶} = {𝐵, 𝐶})
4 preq1b.a . . 3 (𝜑𝐴𝑉)
5 preq1b.b . . 3 (𝜑𝐵𝑊)
64, 5preq1b 4317 . 2 (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵))
73, 6syl5bb 271 1 (𝜑 → ({𝐶, 𝐴} = {𝐶, 𝐵} ↔ 𝐴 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  {cpr 4127 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-sn 4126  df-pr 4128 This theorem is referenced by:  clsk1indlem4  37362  umgr2v2enb1  40742
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