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Theorem clelsb3 2139
Description: Substitution applied to an atomic wff (class version of elsb3 1849). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
Assertion
Ref Expression
clelsb3 ([x / y]y Ax A)
Distinct variable group:   y,A
Allowed substitution hint:   A(x)

Proof of Theorem clelsb3
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 nfv 1418 . . 3 y w A
21sbco2 1836 . 2 ([x / y][y / w]w A ↔ [x / w]w A)
3 nfv 1418 . . . 4 w y A
4 eleq1 2097 . . . 4 (w = y → (w Ay A))
53, 4sbie 1671 . . 3 ([y / w]w Ay A)
65sbbii 1645 . 2 ([x / y][y / w]w A ↔ [x / y]y A)
7 nfv 1418 . . 3 w x A
8 eleq1 2097 . . 3 (w = x → (w Ax A))
97, 8sbie 1671 . 2 ([x / w]w Ax A)
102, 6, 93bitr3i 199 1 ([x / y]y Ax A)
Colors of variables: wff set class
Syntax hints:  wb 98   wcel 1390  [wsb 1642
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-nf 1347  df-sb 1643  df-cleq 2030  df-clel 2033
This theorem is referenced by:  hblem  2142  nfraldya  2352  nfrexdya  2353  cbvreu  2525  sbcel1gv  2815  rmo3  2843  setindel  4221  elirr  4224  en2lp  4232  tfi  4248  bdcriota  9272
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