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Theorem cbvopab2 3822
Description: Change second bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 22-Aug-2013.)
Hypotheses
Ref Expression
cbvopab2.1 zφ
cbvopab2.2 yψ
cbvopab2.3 (y = z → (φψ))
Assertion
Ref Expression
cbvopab2 {⟨x, y⟩ ∣ φ} = {⟨x, z⟩ ∣ ψ}
Distinct variable group:   x,y,z
Allowed substitution hints:   φ(x,y,z)   ψ(x,y,z)

Proof of Theorem cbvopab2
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 nfv 1418 . . . . . 6 z w = ⟨x, y
2 cbvopab2.1 . . . . . 6 zφ
31, 2nfan 1454 . . . . 5 z(w = ⟨x, y φ)
4 nfv 1418 . . . . . 6 y w = ⟨x, z
5 cbvopab2.2 . . . . . 6 yψ
64, 5nfan 1454 . . . . 5 y(w = ⟨x, z ψ)
7 opeq2 3541 . . . . . . 7 (y = z → ⟨x, y⟩ = ⟨x, z⟩)
87eqeq2d 2048 . . . . . 6 (y = z → (w = ⟨x, y⟩ ↔ w = ⟨x, z⟩))
9 cbvopab2.3 . . . . . 6 (y = z → (φψ))
108, 9anbi12d 442 . . . . 5 (y = z → ((w = ⟨x, y φ) ↔ (w = ⟨x, z ψ)))
113, 6, 10cbvex 1636 . . . 4 (y(w = ⟨x, y φ) ↔ z(w = ⟨x, z ψ))
1211exbii 1493 . . 3 (xy(w = ⟨x, y φ) ↔ xz(w = ⟨x, z ψ))
1312abbii 2150 . 2 {wxy(w = ⟨x, y φ)} = {wxz(w = ⟨x, z ψ)}
14 df-opab 3810 . 2 {⟨x, y⟩ ∣ φ} = {wxy(w = ⟨x, y φ)}
15 df-opab 3810 . 2 {⟨x, z⟩ ∣ ψ} = {wxz(w = ⟨x, z ψ)}
1613, 14, 153eqtr4i 2067 1 {⟨x, y⟩ ∣ φ} = {⟨x, 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  {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:  cbvoprab3  5522
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