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Theorem cbvoprab12 5517
Description: Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Hypotheses
Ref Expression
cbvoprab12.1 wφ
cbvoprab12.2 vφ
cbvoprab12.3 xψ
cbvoprab12.4 yψ
cbvoprab12.5 ((x = w y = v) → (φψ))
Assertion
Ref Expression
cbvoprab12 {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨⟨w, v⟩, z⟩ ∣ ψ}
Distinct variable group:   x,y,z,w,v
Allowed substitution hints:   φ(x,y,z,w,v)   ψ(x,y,z,w,v)

Proof of Theorem cbvoprab12
Dummy variable u is distinct from all other variables.
StepHypRef Expression
1 nfv 1418 . . . . 5 w u = ⟨x, y
2 cbvoprab12.1 . . . . 5 wφ
31, 2nfan 1454 . . . 4 w(u = ⟨x, y φ)
4 nfv 1418 . . . . 5 v u = ⟨x, y
5 cbvoprab12.2 . . . . 5 vφ
64, 5nfan 1454 . . . 4 v(u = ⟨x, y φ)
7 nfv 1418 . . . . 5 x u = ⟨w, v
8 cbvoprab12.3 . . . . 5 xψ
97, 8nfan 1454 . . . 4 x(u = ⟨w, v ψ)
10 nfv 1418 . . . . 5 y u = ⟨w, v
11 cbvoprab12.4 . . . . 5 yψ
1210, 11nfan 1454 . . . 4 y(u = ⟨w, v ψ)
13 opeq12 3541 . . . . . 6 ((x = w y = v) → ⟨x, y⟩ = ⟨w, v⟩)
1413eqeq2d 2048 . . . . 5 ((x = w y = v) → (u = ⟨x, y⟩ ↔ u = ⟨w, v⟩))
15 cbvoprab12.5 . . . . 5 ((x = w y = v) → (φψ))
1614, 15anbi12d 442 . . . 4 ((x = w y = v) → ((u = ⟨x, y φ) ↔ (u = ⟨w, v ψ)))
173, 6, 9, 12, 16cbvex2 1794 . . 3 (xy(u = ⟨x, y φ) ↔ wv(u = ⟨w, v ψ))
1817opabbii 3814 . 2 {⟨u, z⟩ ∣ xy(u = ⟨x, y φ)} = {⟨u, z⟩ ∣ wv(u = ⟨w, v ψ)}
19 dfoprab2 5491 . 2 {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨u, z⟩ ∣ xy(u = ⟨x, y φ)}
20 dfoprab2 5491 . 2 {⟨⟨w, v⟩, z⟩ ∣ ψ} = {⟨u, z⟩ ∣ wv(u = ⟨w, v ψ)}
2118, 19, 203eqtr4i 2067 1 {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨⟨w, v⟩, 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:  cbvoprab12v  5518  cbvmpt2x  5521  dfoprab4f  5758  fmpt2x  5765  tposoprab  5833
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