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Theorem cbvopab2v 3825
Description: Rule used to change the second bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 2-Sep-1999.)
Hypothesis
Ref Expression
cbvopab2v.1 (y = z → (φψ))
Assertion
Ref Expression
cbvopab2v {⟨x, y⟩ ∣ φ} = {⟨x, z⟩ ∣ ψ}
Distinct variable groups:   x,y,z   φ,z   ψ,y
Allowed substitution hints:   φ(x,y)   ψ(x,z)

Proof of Theorem cbvopab2v
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 opeq2 3541 . . . . . . 7 (y = z → ⟨x, y⟩ = ⟨x, z⟩)
21eqeq2d 2048 . . . . . 6 (y = z → (w = ⟨x, y⟩ ↔ w = ⟨x, z⟩))
3 cbvopab2v.1 . . . . . 6 (y = z → (φψ))
42, 3anbi12d 442 . . . . 5 (y = z → ((w = ⟨x, y φ) ↔ (w = ⟨x, z ψ)))
54cbvexv 1792 . . . 4 (y(w = ⟨x, y φ) ↔ z(w = ⟨x, z ψ))
65exbii 1493 . . 3 (xy(w = ⟨x, y φ) ↔ xz(w = ⟨x, z ψ))
76abbii 2150 . 2 {wxy(w = ⟨x, y φ)} = {wxz(w = ⟨x, z ψ)}
8 df-opab 3810 . 2 {⟨x, y⟩ ∣ φ} = {wxy(w = ⟨x, y φ)}
9 df-opab 3810 . 2 {⟨x, z⟩ ∣ ψ} = {wxz(w = ⟨x, z ψ)}
107, 8, 93eqtr4i 2067 1 {⟨x, y⟩ ∣ φ} = {⟨x, z⟩ ∣ ψ}
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  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: (None)
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