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Theorem sbabel 2185
 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 1862 . . 3 ([y / x]v(v = {zφ} v A) ↔ v[y / x](v = {zφ} v A))
2 sban 1811 . . . . 5 ([y / x](v = {zφ} v A) ↔ ([y / x]v = {zφ} [y / x]v A))
3 nfv 1402 . . . . . . . . . 10 x z v
43sbf 1642 . . . . . . . . 9 ([y / x]z vz v)
54sbrbis 1817 . . . . . . . 8 ([y / x](z vφ) ↔ (z v ↔ [y / x]φ))
65sbalv 1863 . . . . . . 7 ([y / x]z(z vφ) ↔ z(z v ↔ [y / x]φ))
7 abeq2 2128 . . . . . . . 8 (v = {zφ} ↔ z(z vφ))
87sbbii 1630 . . . . . . 7 ([y / x]v = {zφ} ↔ [y / x]z(z vφ))
9 abeq2 2128 . . . . . . 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 2154 . . . . . . 7 x v A
1312sbf 1642 . . . . . 6 ([y / x]v Av A)
1410, 13anbi12i 436 . . . . 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 1478 . . 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 2018 . . 3 ({zφ} Av(v = {zφ} v A))
1918sbbii 1630 . 2 ([y / x]{zφ} A ↔ [y / x]v(v = {zφ} v A))
20 df-clel 2018 . 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 1226   = wceq 1228  ∃wex 1362   ∈ wcel 1374  [wsb 1627  {cab 2008  Ⅎwnfc 2147 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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149 This theorem is referenced by: (None)
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