MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  preq2b Structured version   Visualization version   GIF version

Theorem preq2b 4351
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 4242 . . 3 {𝐶, 𝐴} = {𝐴, 𝐶}
2 prcom 4242 . . 3 {𝐶, 𝐵} = {𝐵, 𝐶}
31, 2eqeq12i 2640 . 2 ({𝐶, 𝐴} = {𝐶, 𝐵} ↔ {𝐴, 𝐶} = {𝐵, 𝐶})
4 preq1b.a . . 3 (𝜑𝐴𝑉)
5 preq1b.b . . 3 (𝜑𝐵𝑊)
64, 5preq1b 4350 . 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 1480  wcel 1992  {cpr 4155
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 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
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 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-v 3193  df-un 3565  df-sn 4154  df-pr 4156
This theorem is referenced by:  umgr2v2enb1  26302  clsk1indlem4  37810
  Copyright terms: Public domain W3C validator