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Theorem elopab 3969
 Description: Membership in a class abstraction of pairs. (Contributed by NM, 24-Mar-1998.)
Assertion
Ref Expression
elopab (A {⟨x, y⟩ ∣ φ} ↔ xy(A = ⟨x, y φ))
Distinct variable groups:   x,A   y,A
Allowed substitution hints:   φ(x,y)

Proof of Theorem elopab
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 elex 2543 . 2 (A {⟨x, y⟩ ∣ φ} → A V)
2 vex 2538 . . . . . 6 x V
3 vex 2538 . . . . . 6 y V
42, 3opex 3940 . . . . 5 x, y V
5 eleq1 2082 . . . . 5 (A = ⟨x, y⟩ → (A V ↔ ⟨x, y V))
64, 5mpbiri 157 . . . 4 (A = ⟨x, y⟩ → A V)
76adantr 261 . . 3 ((A = ⟨x, y φ) → A V)
87exlimivv 1758 . 2 (xy(A = ⟨x, y φ) → A V)
9 eqeq1 2028 . . . . 5 (z = A → (z = ⟨x, y⟩ ↔ A = ⟨x, y⟩))
109anbi1d 441 . . . 4 (z = A → ((z = ⟨x, y φ) ↔ (A = ⟨x, y φ)))
11102exbidv 1730 . . 3 (z = A → (xy(z = ⟨x, y φ) ↔ xy(A = ⟨x, y φ)))
12 df-opab 3793 . . 3 {⟨x, y⟩ ∣ φ} = {zxy(z = ⟨x, y φ)}
1311, 12elab2g 2666 . 2 (A V → (A {⟨x, y⟩ ∣ φ} ↔ xy(A = ⟨x, y φ)))
141, 8, 13pm5.21nii 607 1 (A {⟨x, y⟩ ∣ φ} ↔ xy(A = ⟨x, y φ))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1228  ∃wex 1362   ∈ wcel 1374  Vcvv 2535  ⟨cop 3353  {copab 3791 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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-opab 3793 This theorem is referenced by:  opelopabsbALT  3970  opelopabsb  3971  opelopabt  3973  opelopabga  3974  opabm  3991  iunopab  3992  epelg  4001  elxp  4289  elcnv  4439  dfmpt3  4947  0neqopab  5473  brabvv  5474  opabex3d  5671  opabex3  5672
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