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Theorem clelsb3 2455
 Description: Substitution applied to an atomic wff (class version of elsb3 2103). (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 1619 . . 3 y w A
21sbco2 2086 . 2 ([x / y][y / w]w A ↔ [x / w]w A)
3 nfv 1619 . . . 4 w y A
4 eleq1 2413 . . . 4 (w = y → (w Ay A))
53, 4sbie 2038 . . 3 ([y / w]w Ay A)
65sbbii 1653 . 2 ([x / y][y / w]w A ↔ [x / y]y A)
7 nfv 1619 . . 3 w x A
8 eleq1 2413 . . 3 (w = x → (w Ax A))
97, 8sbie 2038 . 2 ([x / w]w Ax A)
102, 6, 93bitr3i 266 1 ([x / y]y Ax A)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176  [wsb 1648   ∈ wcel 1710 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349 This theorem is referenced by:  hblem  2457  cbvreu  2833  sbcel1gv  3105  rmo3  3133
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