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Theorem cbvoprab1 5515
Description: Rule used to change first bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 5-Dec-2016.)
Hypotheses
Ref Expression
cbvoprab1.1 wφ
cbvoprab1.2 xψ
cbvoprab1.3 (x = w → (φψ))
Assertion
Ref Expression
cbvoprab1 {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨⟨w, y⟩, z⟩ ∣ ψ}
Distinct variable group:   x,y,z,w
Allowed substitution hints:   φ(x,y,z,w)   ψ(x,y,z,w)

Proof of Theorem cbvoprab1
Dummy variable v is distinct from all other variables.
StepHypRef Expression
1 nfv 1418 . . . . . 6 w v = ⟨x, y
2 cbvoprab1.1 . . . . . 6 wφ
31, 2nfan 1454 . . . . 5 w(v = ⟨x, y φ)
43nfex 1525 . . . 4 wy(v = ⟨x, y φ)
5 nfv 1418 . . . . . 6 x v = ⟨w, y
6 cbvoprab1.2 . . . . . 6 xψ
75, 6nfan 1454 . . . . 5 x(v = ⟨w, y ψ)
87nfex 1525 . . . 4 xy(v = ⟨w, y ψ)
9 opeq1 3539 . . . . . . 7 (x = w → ⟨x, y⟩ = ⟨w, y⟩)
109eqeq2d 2048 . . . . . 6 (x = w → (v = ⟨x, y⟩ ↔ v = ⟨w, y⟩))
11 cbvoprab1.3 . . . . . 6 (x = w → (φψ))
1210, 11anbi12d 442 . . . . 5 (x = w → ((v = ⟨x, y φ) ↔ (v = ⟨w, y ψ)))
1312exbidv 1703 . . . 4 (x = w → (y(v = ⟨x, y φ) ↔ y(v = ⟨w, y ψ)))
144, 8, 13cbvex 1636 . . 3 (xy(v = ⟨x, y φ) ↔ wy(v = ⟨w, y ψ))
1514opabbii 3814 . 2 {⟨v, z⟩ ∣ xy(v = ⟨x, y φ)} = {⟨v, z⟩ ∣ wy(v = ⟨w, y ψ)}
16 dfoprab2 5491 . 2 {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨v, z⟩ ∣ xy(v = ⟨x, y φ)}
17 dfoprab2 5491 . 2 {⟨⟨w, y⟩, z⟩ ∣ ψ} = {⟨v, z⟩ ∣ wy(v = ⟨w, y ψ)}
1815, 16, 173eqtr4i 2067 1 {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨⟨w, y⟩, z⟩ ∣ ψ}
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wnf 1346  wex 1378  cop 3369  {copab 3807  {coprab 5453
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 3865  ax-pow 3917  ax-pr 3934
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-in 2918  df-ss 2925  df-pw 3352  df-sn 3372  df-pr 3373  df-op 3375  df-opab 3809  df-oprab 5456
This theorem is referenced by: (None)
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