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

Theorem sbceqg 2860
 Description: Distribute proper substitution through an equality relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
sbceqg

Proof of Theorem sbceqg
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 2761 . . 3
2 dfsbcq2 2761 . . . . 5
32abbidv 2152 . . . 4
4 dfsbcq2 2761 . . . . 5
54abbidv 2152 . . . 4
63, 5eqeq12d 2051 . . 3
7 nfs1v 1812 . . . . . 6
87nfab 2179 . . . . 5
9 nfs1v 1812 . . . . . 6
109nfab 2179 . . . . 5
118, 10nfeq 2182 . . . 4
12 sbab 2161 . . . . 5
13 sbab 2161 . . . . 5
1412, 13eqeq12d 2051 . . . 4
1511, 14sbie 1671 . . 3
161, 6, 15vtoclbg 2608 . 2
17 df-csb 2847 . . 3
18 df-csb 2847 . . 3
1917, 18eqeq12i 2050 . 2
2016, 19syl6bbr 187 1
 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:  sbcne12g  2862  sbceq1g  2864  sbceq2g  2866  sbcfng  4987
 Copyright terms: Public domain W3C validator