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Theorem cnvopab 4669
Description: The converse of a class abstraction of ordered pairs. (Contributed by NM, 11-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
cnvopab {⟨x, y⟩ ∣ φ} = {⟨y, x⟩ ∣ φ}
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem cnvopab
Dummy variables z w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relcnv 4646 . 2 Rel {⟨x, y⟩ ∣ φ}
2 relopab 4407 . 2 Rel {⟨y, x⟩ ∣ φ}
3 opelopabsbALT 3987 . . . 4 (⟨w, z {⟨x, y⟩ ∣ φ} ↔ [z / y][w / x]φ)
4 sbcom2 1860 . . . 4 ([z / y][w / x]φ ↔ [w / x][z / y]φ)
53, 4bitri 173 . . 3 (⟨w, z {⟨x, y⟩ ∣ φ} ↔ [w / x][z / y]φ)
6 vex 2554 . . . 4 z V
7 vex 2554 . . . 4 w V
86, 7opelcnv 4460 . . 3 (⟨z, w {⟨x, y⟩ ∣ φ} ↔ ⟨w, z {⟨x, y⟩ ∣ φ})
9 opelopabsbALT 3987 . . 3 (⟨z, w {⟨y, x⟩ ∣ φ} ↔ [w / x][z / y]φ)
105, 8, 93bitr4i 201 . 2 (⟨z, w {⟨x, y⟩ ∣ φ} ↔ ⟨z, w {⟨y, x⟩ ∣ φ})
111, 2, 10eqrelriiv 4377 1 {⟨x, y⟩ ∣ φ} = {⟨y, x⟩ ∣ φ}
Colors of variables: wff set class
Syntax hints:   = wceq 1242   wcel 1390  [wsb 1642  cop 3370  {copab 3808  ccnv 4287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296
This theorem is referenced by:  cnvxp  4685  mptpreima  4757  f1ocnvd  5644
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