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Theorem sbcabel 2833
Description: Interchange class substitution and class abstraction. (Contributed by NM, 5-Nov-2005.)
Hypothesis
Ref Expression
sbcabel.1 xB
Assertion
Ref Expression
sbcabel (A 𝑉 → ([A / x]{yφ} B ↔ {y[A / x]φ} B))
Distinct variable groups:   y,A   x,y
Allowed substitution hints:   φ(x,y)   A(x)   B(x,y)   𝑉(x,y)

Proof of Theorem sbcabel
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 elex 2560 . 2 (A 𝑉A V)
2 sbcexg 2807 . . . 4 (A V → ([A / x]w(w = {yφ} w B) ↔ w[A / x](w = {yφ} w B)))
3 sbcang 2800 . . . . . 6 (A V → ([A / x](w = {yφ} w B) ↔ ([A / x]w = {yφ} [A / x]w B)))
4 sbcalg 2805 . . . . . . . . 9 (A V → ([A / x]y(y wφ) ↔ y[A / x](y wφ)))
5 sbcbig 2803 . . . . . . . . . . 11 (A V → ([A / x](y wφ) ↔ ([A / x]y w[A / x]φ)))
6 sbcg 2821 . . . . . . . . . . . 12 (A V → ([A / x]y wy w))
76bibi1d 222 . . . . . . . . . . 11 (A V → (([A / x]y w[A / x]φ) ↔ (y w[A / x]φ)))
85, 7bitrd 177 . . . . . . . . . 10 (A V → ([A / x](y wφ) ↔ (y w[A / x]φ)))
98albidv 1702 . . . . . . . . 9 (A V → (y[A / x](y wφ) ↔ y(y w[A / x]φ)))
104, 9bitrd 177 . . . . . . . 8 (A V → ([A / x]y(y wφ) ↔ y(y w[A / x]φ)))
11 abeq2 2143 . . . . . . . . 9 (w = {yφ} ↔ y(y wφ))
1211sbcbii 2812 . . . . . . . 8 ([A / x]w = {yφ} ↔ [A / x]y(y wφ))
13 abeq2 2143 . . . . . . . 8 (w = {y[A / x]φ} ↔ y(y w[A / x]φ))
1410, 12, 133bitr4g 212 . . . . . . 7 (A V → ([A / x]w = {yφ} ↔ w = {y[A / x]φ}))
15 sbcabel.1 . . . . . . . . 9 xB
1615nfcri 2169 . . . . . . . 8 x w B
1716sbcgf 2819 . . . . . . 7 (A V → ([A / x]w Bw B))
1814, 17anbi12d 442 . . . . . 6 (A V → (([A / x]w = {yφ} [A / x]w B) ↔ (w = {y[A / x]φ} w B)))
193, 18bitrd 177 . . . . 5 (A V → ([A / x](w = {yφ} w B) ↔ (w = {y[A / x]φ} w B)))
2019exbidv 1703 . . . 4 (A V → (w[A / x](w = {yφ} w B) ↔ w(w = {y[A / x]φ} w B)))
212, 20bitrd 177 . . 3 (A V → ([A / x]w(w = {yφ} w B) ↔ w(w = {y[A / x]φ} w B)))
22 df-clel 2033 . . . 4 ({yφ} Bw(w = {yφ} w B))
2322sbcbii 2812 . . 3 ([A / x]{yφ} B[A / x]w(w = {yφ} w B))
24 df-clel 2033 . . 3 ({y[A / x]φ} Bw(w = {y[A / x]φ} w B))
2521, 23, 243bitr4g 212 . 2 (A V → ([A / x]{yφ} B ↔ {y[A / x]φ} B))
261, 25syl 14 1 (A 𝑉 → ([A / x]{yφ} B ↔ {y[A / x]φ} B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   = wceq 1242  wex 1378   wcel 1390  {cab 2023  wnfc 2162  Vcvv 2551  [wsbc 2758
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-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759
This theorem is referenced by:  csbexga  3876
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