ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbcabel Structured version   GIF version

Theorem sbcabel 2816
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 2543 . 2 (A 𝑉A V)
2 sbcexg 2790 . . . 4 (A V → ([A / x]w(w = {yφ} w B) ↔ w[A / x](w = {yφ} w B)))
3 sbcang 2783 . . . . . 6 (A V → ([A / x](w = {yφ} w B) ↔ ([A / x]w = {yφ} [A / x]w B)))
4 sbcalg 2788 . . . . . . . . 9 (A V → ([A / x]y(y wφ) ↔ y[A / x](y wφ)))
5 sbcbig 2786 . . . . . . . . . . 11 (A V → ([A / x](y wφ) ↔ ([A / x]y w[A / x]φ)))
6 sbcg 2804 . . . . . . . . . . . 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 1687 . . . . . . . . 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 2128 . . . . . . . . 9 (w = {yφ} ↔ y(y wφ))
1211sbcbii 2795 . . . . . . . 8 ([A / x]w = {yφ} ↔ [A / x]y(y wφ))
13 abeq2 2128 . . . . . . . 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 2154 . . . . . . . 8 x w B
1716sbcgf 2802 . . . . . . 7 (A V → ([A / x]w Bw B))
1814, 17anbi12d 445 . . . . . 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 1688 . . . 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 2018 . . . 4 ({yφ} Bw(w = {yφ} w B))
2322sbcbii 2795 . . 3 ([A / x]{yφ} B[A / x]w(w = {yφ} w B))
24 df-clel 2018 . . 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 1226   = wceq 1228  wex 1362   wcel 1374  {cab 2008  wnfc 2147  Vcvv 2535  [wsbc 2741
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-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-sbc 2742
This theorem is referenced by:  csbexgOLD  3859
  Copyright terms: Public domain W3C validator