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Theorem cbvoprab2 5519
Description: Change the second bound variable in an operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)
Hypotheses
Ref Expression
cbvoprab2.1 wφ
cbvoprab2.2 yψ
cbvoprab2.3 (y = w → (φψ))
Assertion
Ref Expression
cbvoprab2 {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨⟨x, w⟩, z⟩ ∣ ψ}
Distinct variable group:   x,w,y,z
Allowed substitution hints:   φ(x,y,z,w)   ψ(x,y,z,w)

Proof of Theorem cbvoprab2
Dummy variable v is distinct from all other variables.
StepHypRef Expression
1 nfv 1418 . . . . . . 7 w v = ⟨⟨x, y⟩, z
2 cbvoprab2.1 . . . . . . 7 wφ
31, 2nfan 1454 . . . . . 6 w(v = ⟨⟨x, y⟩, z φ)
43nfex 1525 . . . . 5 wz(v = ⟨⟨x, y⟩, z φ)
5 nfv 1418 . . . . . . 7 y v = ⟨⟨x, w⟩, z
6 cbvoprab2.2 . . . . . . 7 yψ
75, 6nfan 1454 . . . . . 6 y(v = ⟨⟨x, w⟩, z ψ)
87nfex 1525 . . . . 5 yz(v = ⟨⟨x, w⟩, z ψ)
9 opeq2 3541 . . . . . . . . 9 (y = w → ⟨x, y⟩ = ⟨x, w⟩)
109opeq1d 3546 . . . . . . . 8 (y = w → ⟨⟨x, y⟩, z⟩ = ⟨⟨x, w⟩, z⟩)
1110eqeq2d 2048 . . . . . . 7 (y = w → (v = ⟨⟨x, y⟩, z⟩ ↔ v = ⟨⟨x, w⟩, z⟩))
12 cbvoprab2.3 . . . . . . 7 (y = w → (φψ))
1311, 12anbi12d 442 . . . . . 6 (y = w → ((v = ⟨⟨x, y⟩, z φ) ↔ (v = ⟨⟨x, w⟩, z ψ)))
1413exbidv 1703 . . . . 5 (y = w → (z(v = ⟨⟨x, y⟩, z φ) ↔ z(v = ⟨⟨x, w⟩, z ψ)))
154, 8, 14cbvex 1636 . . . 4 (yz(v = ⟨⟨x, y⟩, z φ) ↔ wz(v = ⟨⟨x, w⟩, z ψ))
1615exbii 1493 . . 3 (xyz(v = ⟨⟨x, y⟩, z φ) ↔ xwz(v = ⟨⟨x, w⟩, z ψ))
1716abbii 2150 . 2 {vxyz(v = ⟨⟨x, y⟩, z φ)} = {vxwz(v = ⟨⟨x, w⟩, z ψ)}
18 df-oprab 5459 . 2 {⟨⟨x, y⟩, z⟩ ∣ φ} = {vxyz(v = ⟨⟨x, y⟩, z φ)}
19 df-oprab 5459 . 2 {⟨⟨x, w⟩, z⟩ ∣ ψ} = {vxwz(v = ⟨⟨x, w⟩, z ψ)}
2017, 18, 193eqtr4i 2067 1 {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨⟨x, w⟩, z⟩ ∣ 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  {coprab 5456
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-oprab 5459
This theorem is referenced by: (None)
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