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Theorem cbvopab1s 3823
Description: Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 31-Jul-2003.)
Assertion
Ref Expression
cbvopab1s {⟨x, y⟩ ∣ φ} = {⟨z, y⟩ ∣ [z / x]φ}
Distinct variable groups:   x,y,z   φ,z
Allowed substitution hints:   φ(x,y)

Proof of Theorem cbvopab1s
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 nfv 1418 . . . 4 zy(w = ⟨x, y φ)
2 nfv 1418 . . . . . 6 x w = ⟨z, y
3 nfs1v 1812 . . . . . 6 x[z / x]φ
42, 3nfan 1454 . . . . 5 x(w = ⟨z, y [z / x]φ)
54nfex 1525 . . . 4 xy(w = ⟨z, y [z / x]φ)
6 opeq1 3540 . . . . . . 7 (x = z → ⟨x, y⟩ = ⟨z, y⟩)
76eqeq2d 2048 . . . . . 6 (x = z → (w = ⟨x, y⟩ ↔ w = ⟨z, y⟩))
8 sbequ12 1651 . . . . . 6 (x = z → (φ ↔ [z / x]φ))
97, 8anbi12d 442 . . . . 5 (x = z → ((w = ⟨x, y φ) ↔ (w = ⟨z, y [z / x]φ)))
109exbidv 1703 . . . 4 (x = z → (y(w = ⟨x, y φ) ↔ y(w = ⟨z, y [z / x]φ)))
111, 5, 10cbvex 1636 . . 3 (xy(w = ⟨x, y φ) ↔ zy(w = ⟨z, y [z / x]φ))
1211abbii 2150 . 2 {wxy(w = ⟨x, y φ)} = {wzy(w = ⟨z, y [z / x]φ)}
13 df-opab 3810 . 2 {⟨x, y⟩ ∣ φ} = {wxy(w = ⟨x, y φ)}
14 df-opab 3810 . 2 {⟨z, y⟩ ∣ [z / x]φ} = {wzy(w = ⟨z, y [z / x]φ)}
1512, 13, 143eqtr4i 2067 1 {⟨x, y⟩ ∣ φ} = {⟨z, y⟩ ∣ [z / x]φ}
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242  wex 1378  [wsb 1642  {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: (None)
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