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Theorem cbvopab1 3820
Description: Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 6-Oct-2004.) (Revised by Mario Carneiro, 14-Oct-2016.)
Hypotheses
Ref Expression
cbvopab1.1 zφ
cbvopab1.2 xψ
cbvopab1.3 (x = z → (φψ))
Assertion
Ref Expression
cbvopab1 {⟨x, y⟩ ∣ φ} = {⟨z, y⟩ ∣ ψ}
Distinct variable groups:   x,y   y,z
Allowed substitution hints:   φ(x,y,z)   ψ(x,y,z)

Proof of Theorem cbvopab1
Dummy variables w v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1418 . . . . 5 vy(w = ⟨x, y φ)
2 nfv 1418 . . . . . . 7 x w = ⟨v, y
3 nfs1v 1812 . . . . . . 7 x[v / x]φ
42, 3nfan 1454 . . . . . 6 x(w = ⟨v, y [v / x]φ)
54nfex 1525 . . . . 5 xy(w = ⟨v, y [v / x]φ)
6 opeq1 3539 . . . . . . . 8 (x = v → ⟨x, y⟩ = ⟨v, y⟩)
76eqeq2d 2048 . . . . . . 7 (x = v → (w = ⟨x, y⟩ ↔ w = ⟨v, y⟩))
8 sbequ12 1651 . . . . . . 7 (x = v → (φ ↔ [v / x]φ))
97, 8anbi12d 442 . . . . . 6 (x = v → ((w = ⟨x, y φ) ↔ (w = ⟨v, y [v / x]φ)))
109exbidv 1703 . . . . 5 (x = v → (y(w = ⟨x, y φ) ↔ y(w = ⟨v, y [v / x]φ)))
111, 5, 10cbvex 1636 . . . 4 (xy(w = ⟨x, y φ) ↔ vy(w = ⟨v, y [v / x]φ))
12 nfv 1418 . . . . . . 7 z w = ⟨v, y
13 cbvopab1.1 . . . . . . . 8 zφ
1413nfsb 1819 . . . . . . 7 z[v / x]φ
1512, 14nfan 1454 . . . . . 6 z(w = ⟨v, y [v / x]φ)
1615nfex 1525 . . . . 5 zy(w = ⟨v, y [v / x]φ)
17 nfv 1418 . . . . 5 vy(w = ⟨z, y ψ)
18 opeq1 3539 . . . . . . . 8 (v = z → ⟨v, y⟩ = ⟨z, y⟩)
1918eqeq2d 2048 . . . . . . 7 (v = z → (w = ⟨v, y⟩ ↔ w = ⟨z, y⟩))
20 sbequ 1718 . . . . . . . 8 (v = z → ([v / x]φ ↔ [z / x]φ))
21 cbvopab1.2 . . . . . . . . 9 xψ
22 cbvopab1.3 . . . . . . . . 9 (x = z → (φψ))
2321, 22sbie 1671 . . . . . . . 8 ([z / x]φψ)
2420, 23syl6bb 185 . . . . . . 7 (v = z → ([v / x]φψ))
2519, 24anbi12d 442 . . . . . 6 (v = z → ((w = ⟨v, y [v / x]φ) ↔ (w = ⟨z, y ψ)))
2625exbidv 1703 . . . . 5 (v = z → (y(w = ⟨v, y [v / x]φ) ↔ y(w = ⟨z, y ψ)))
2716, 17, 26cbvex 1636 . . . 4 (vy(w = ⟨v, y [v / x]φ) ↔ zy(w = ⟨z, y ψ))
2811, 27bitri 173 . . 3 (xy(w = ⟨x, y φ) ↔ zy(w = ⟨z, y ψ))
2928abbii 2150 . 2 {wxy(w = ⟨x, y φ)} = {wzy(w = ⟨z, y ψ)}
30 df-opab 3809 . 2 {⟨x, y⟩ ∣ φ} = {wxy(w = ⟨x, y φ)}
31 df-opab 3809 . 2 {⟨z, y⟩ ∣ ψ} = {wzy(w = ⟨z, y ψ)}
3229, 30, 313eqtr4i 2067 1 {⟨x, y⟩ ∣ φ} = {⟨z, y⟩ ∣ ψ}
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wnf 1346  wex 1378  [wsb 1642  {cab 2023  cop 3369  {copab 3807
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 3372  df-pr 3373  df-op 3375  df-opab 3809
This theorem is referenced by:  cbvopab1v  3823  cbvmpt  3841
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