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Theorem cbvopab 3819
Description: Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 14-Sep-2003.)
Hypotheses
Ref Expression
cbvopab.1 zφ
cbvopab.2 wφ
cbvopab.3 xψ
cbvopab.4 yψ
cbvopab.5 ((x = z y = w) → (φψ))
Assertion
Ref Expression
cbvopab {⟨x, y⟩ ∣ φ} = {⟨z, w⟩ ∣ ψ}
Distinct variable group:   x,y,z,w
Allowed substitution hints:   φ(x,y,z,w)   ψ(x,y,z,w)

Proof of Theorem cbvopab
Dummy variable v is distinct from all other variables.
StepHypRef Expression
1 nfv 1418 . . . . 5 z v = ⟨x, y
2 cbvopab.1 . . . . 5 zφ
31, 2nfan 1454 . . . 4 z(v = ⟨x, y φ)
4 nfv 1418 . . . . 5 w v = ⟨x, y
5 cbvopab.2 . . . . 5 wφ
64, 5nfan 1454 . . . 4 w(v = ⟨x, y φ)
7 nfv 1418 . . . . 5 x v = ⟨z, w
8 cbvopab.3 . . . . 5 xψ
97, 8nfan 1454 . . . 4 x(v = ⟨z, w ψ)
10 nfv 1418 . . . . 5 y v = ⟨z, w
11 cbvopab.4 . . . . 5 yψ
1210, 11nfan 1454 . . . 4 y(v = ⟨z, w ψ)
13 opeq12 3542 . . . . . 6 ((x = z y = w) → ⟨x, y⟩ = ⟨z, w⟩)
1413eqeq2d 2048 . . . . 5 ((x = z y = w) → (v = ⟨x, y⟩ ↔ v = ⟨z, w⟩))
15 cbvopab.5 . . . . 5 ((x = z y = w) → (φψ))
1614, 15anbi12d 442 . . . 4 ((x = z y = w) → ((v = ⟨x, y φ) ↔ (v = ⟨z, w ψ)))
173, 6, 9, 12, 16cbvex2 1794 . . 3 (xy(v = ⟨x, y φ) ↔ zw(v = ⟨z, w ψ))
1817abbii 2150 . 2 {vxy(v = ⟨x, y φ)} = {vzw(v = ⟨z, w ψ)}
19 df-opab 3810 . 2 {⟨x, y⟩ ∣ φ} = {vxy(v = ⟨x, y φ)}
20 df-opab 3810 . 2 {⟨z, w⟩ ∣ ψ} = {vzw(v = ⟨z, w ψ)}
2118, 19, 203eqtr4i 2067 1 {⟨x, y⟩ ∣ φ} = {⟨z, w⟩ ∣ ψ}
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wnf 1346  wex 1378  {cab 2023  cop 3370  {copab 3808
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-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-opab 3810
This theorem is referenced by:  cbvopabv  3820  opelopabsb  3988
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