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Theorem sbcel12g 2859
Description: Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
sbcel12g (A 𝑉 → ([A / x]B 𝐶A / xB A / x𝐶))

Proof of Theorem sbcel12g
Dummy variables y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 2761 . . 3 (z = A → ([z / x]B 𝐶[A / x]B 𝐶))
2 dfsbcq2 2761 . . . . 5 (z = A → ([z / x]y B[A / x]y B))
32abbidv 2152 . . . 4 (z = A → {y ∣ [z / x]y B} = {y[A / x]y B})
4 dfsbcq2 2761 . . . . 5 (z = A → ([z / x]y 𝐶[A / x]y 𝐶))
54abbidv 2152 . . . 4 (z = A → {y ∣ [z / x]y 𝐶} = {y[A / x]y 𝐶})
63, 5eleq12d 2105 . . 3 (z = A → ({y ∣ [z / x]y B} {y ∣ [z / x]y 𝐶} ↔ {y[A / x]y B} {y[A / x]y 𝐶}))
7 nfs1v 1812 . . . . . 6 x[z / x]y B
87nfab 2179 . . . . 5 x{y ∣ [z / x]y B}
9 nfs1v 1812 . . . . . 6 x[z / x]y 𝐶
109nfab 2179 . . . . 5 x{y ∣ [z / x]y 𝐶}
118, 10nfel 2183 . . . 4 x{y ∣ [z / x]y B} {y ∣ [z / x]y 𝐶}
12 sbab 2161 . . . . 5 (x = zB = {y ∣ [z / x]y B})
13 sbab 2161 . . . . 5 (x = z𝐶 = {y ∣ [z / x]y 𝐶})
1412, 13eleq12d 2105 . . . 4 (x = z → (B 𝐶 ↔ {y ∣ [z / x]y B} {y ∣ [z / x]y 𝐶}))
1511, 14sbie 1671 . . 3 ([z / x]B 𝐶 ↔ {y ∣ [z / x]y B} {y ∣ [z / x]y 𝐶})
161, 6, 15vtoclbg 2608 . 2 (A 𝑉 → ([A / x]B 𝐶 ↔ {y[A / x]y B} {y[A / x]y 𝐶}))
17 df-csb 2847 . . 3 A / xB = {y[A / x]y B}
18 df-csb 2847 . . 3 A / x𝐶 = {y[A / x]y 𝐶}
1917, 18eleq12i 2102 . 2 (A / xB A / x𝐶 ↔ {y[A / x]y B} {y[A / x]y 𝐶})
2016, 19syl6bbr 187 1 (A 𝑉 → ([A / x]B 𝐶A / xB A / x𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242   wcel 1390  [wsb 1642  {cab 2023  [wsbc 2758  csb 2846
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  df-csb 2847
This theorem is referenced by:  sbcnel12g  2861  sbcel1g  2863  sbcel2g  2865  sbccsb2g  2873
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