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Theorem sbabel 2200
Description: Theorem to move a substitution in and out of a class abstraction. (Contributed by NM, 27-Sep-2003.) (Revised by Mario Carneiro, 7-Oct-2016.)
Hypothesis
Ref Expression
sbabel.1 xA
Assertion
Ref Expression
sbabel ([y / x]{zφ} A ↔ {z ∣ [y / x]φ} A)
Distinct variable groups:   x,z   y,z
Allowed substitution hints:   φ(x,y,z)   A(x,y,z)

Proof of Theorem sbabel
Dummy variable v is distinct from all other variables.
StepHypRef Expression
1 sbex 1877 . . 3 ([y / x]v(v = {zφ} v A) ↔ v[y / x](v = {zφ} v A))
2 sban 1826 . . . . 5 ([y / x](v = {zφ} v A) ↔ ([y / x]v = {zφ} [y / x]v A))
3 nfv 1418 . . . . . . . . . 10 x z v
43sbf 1657 . . . . . . . . 9 ([y / x]z vz v)
54sbrbis 1832 . . . . . . . 8 ([y / x](z vφ) ↔ (z v ↔ [y / x]φ))
65sbalv 1878 . . . . . . 7 ([y / x]z(z vφ) ↔ z(z v ↔ [y / x]φ))
7 abeq2 2143 . . . . . . . 8 (v = {zφ} ↔ z(z vφ))
87sbbii 1645 . . . . . . 7 ([y / x]v = {zφ} ↔ [y / x]z(z vφ))
9 abeq2 2143 . . . . . . 7 (v = {z ∣ [y / x]φ} ↔ z(z v ↔ [y / x]φ))
106, 8, 93bitr4i 201 . . . . . 6 ([y / x]v = {zφ} ↔ v = {z ∣ [y / x]φ})
11 sbabel.1 . . . . . . . 8 xA
1211nfcri 2169 . . . . . . 7 x v A
1312sbf 1657 . . . . . 6 ([y / x]v Av A)
1410, 13anbi12i 433 . . . . 5 (([y / x]v = {zφ} [y / x]v A) ↔ (v = {z ∣ [y / x]φ} v A))
152, 14bitri 173 . . . 4 ([y / x](v = {zφ} v A) ↔ (v = {z ∣ [y / x]φ} v A))
1615exbii 1493 . . 3 (v[y / x](v = {zφ} v A) ↔ v(v = {z ∣ [y / x]φ} v A))
171, 16bitri 173 . 2 ([y / x]v(v = {zφ} v A) ↔ v(v = {z ∣ [y / x]φ} v A))
18 df-clel 2033 . . 3 ({zφ} Av(v = {zφ} v A))
1918sbbii 1645 . 2 ([y / x]{zφ} A ↔ [y / x]v(v = {zφ} v A))
20 df-clel 2033 . 2 ({z ∣ [y / x]φ} Av(v = {z ∣ [y / x]φ} v A))
2117, 19, 203bitr4i 201 1 ([y / x]{zφ} A ↔ {z ∣ [y / x]φ} A)
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98  wal 1240   = wceq 1242  wex 1378   wcel 1390  [wsb 1642  {cab 2023  wnfc 2162
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-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164
This theorem is referenced by: (None)
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