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Theorem cnvopabOLD 6161
Description: Obsolete version of cnvopab 6160 as of 7-Jun-2025. (Contributed by NM, 11-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
cnvopabOLD {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑦, 𝑥⟩ ∣ 𝜑}
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem cnvopabOLD
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relcnv 6125 . 2 Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2 relopabv 5834 . 2 Rel {⟨𝑦, 𝑥⟩ ∣ 𝜑}
3 vopelopabsb 5539 . . . 4 (⟨𝑤, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑥][𝑧 / 𝑦]𝜑)
4 sbcom2 2171 . . . 4 ([𝑤 / 𝑥][𝑧 / 𝑦]𝜑 ↔ [𝑧 / 𝑦][𝑤 / 𝑥]𝜑)
53, 4bitri 275 . . 3 (⟨𝑤, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑦][𝑤 / 𝑥]𝜑)
6 vex 3482 . . . 4 𝑧 ∈ V
7 vex 3482 . . . 4 𝑤 ∈ V
86, 7opelcnv 5895 . . 3 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
9 vopelopabsb 5539 . . 3 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑦, 𝑥⟩ ∣ 𝜑} ↔ [𝑧 / 𝑦][𝑤 / 𝑥]𝜑)
105, 8, 93bitr4i 303 . 2 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑦, 𝑥⟩ ∣ 𝜑})
111, 2, 10eqrelriiv 5803 1 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑦, 𝑥⟩ ∣ 𝜑}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  [wsb 2062  wcel 2106  cop 4637  {copab 5210  ccnv 5688
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-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697
This theorem is referenced by: (None)
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