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Theorem cbvoprab3 5519
 Description: Rule used to change the third bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 22-Aug-2013.)
Hypotheses
Ref Expression
cbvoprab3.1 wφ
cbvoprab3.2 zψ
cbvoprab3.3 (z = w → (φψ))
Assertion
Ref Expression
cbvoprab3 {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨⟨x, y⟩, w⟩ ∣ ψ}
Distinct variable groups:   x,z,w   y,z,w
Allowed substitution hints:   φ(x,y,z,w)   ψ(x,y,z,w)

Proof of Theorem cbvoprab3
Dummy variable v is distinct from all other variables.
StepHypRef Expression
1 nfv 1418 . . . . . 6 w v = ⟨x, y
2 cbvoprab3.1 . . . . . 6 wφ
31, 2nfan 1454 . . . . 5 w(v = ⟨x, y φ)
43nfex 1525 . . . 4 wy(v = ⟨x, y φ)
54nfex 1525 . . 3 wxy(v = ⟨x, y φ)
6 nfv 1418 . . . . . 6 z v = ⟨x, y
7 cbvoprab3.2 . . . . . 6 zψ
86, 7nfan 1454 . . . . 5 z(v = ⟨x, y ψ)
98nfex 1525 . . . 4 zy(v = ⟨x, y ψ)
109nfex 1525 . . 3 zxy(v = ⟨x, y ψ)
11 cbvoprab3.3 . . . . 5 (z = w → (φψ))
1211anbi2d 437 . . . 4 (z = w → ((v = ⟨x, y φ) ↔ (v = ⟨x, y ψ)))
13122exbidv 1745 . . 3 (z = w → (xy(v = ⟨x, y φ) ↔ xy(v = ⟨x, y ψ)))
145, 10, 13cbvopab2 3821 . 2 {⟨v, z⟩ ∣ xy(v = ⟨x, y φ)} = {⟨v, w⟩ ∣ xy(v = ⟨x, y ψ)}
15 dfoprab2 5491 . 2 {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨v, z⟩ ∣ xy(v = ⟨x, y φ)}
16 dfoprab2 5491 . 2 {⟨⟨x, y⟩, w⟩ ∣ ψ} = {⟨v, w⟩ ∣ xy(v = ⟨x, y ψ)}
1714, 15, 163eqtr4i 2067 1 {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨⟨x, y⟩, w⟩ ∣ ψ}
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242  Ⅎwnf 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:  cbvoprab3v  5520  tposoprab  5833  erovlem  6127
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