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Theorem sbceqg 2866
 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 2767 . . 3
2 dfsbcq2 2767 . . . . 5
32abbidv 2155 . . . 4
4 dfsbcq2 2767 . . . . 5
54abbidv 2155 . . . 4
63, 5eqeq12d 2054 . . 3
7 nfs1v 1815 . . . . . 6
87nfab 2182 . . . . 5
9 nfs1v 1815 . . . . . 6
109nfab 2182 . . . . 5
118, 10nfeq 2185 . . . 4
12 sbab 2164 . . . . 5
13 sbab 2164 . . . . 5
1412, 13eqeq12d 2054 . . . 4
1511, 14sbie 1674 . . 3
161, 6, 15vtoclbg 2614 . 2
17 df-csb 2853 . . 3
18 df-csb 2853 . . 3
1917, 18eqeq12i 2053 . 2
2016, 19syl6bbr 187 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 98   wceq 1243   wcel 1393  wsb 1645  cab 2026  wsbc 2764  csb 2852 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765  df-csb 2853 This theorem is referenced by:  sbcne12g  2868  sbceq1g  2870  sbceq2g  2872  sbcfng  5044
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