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Theorem elsb4 2104
 Description: Substitution applied to an atomic membership wff. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
Assertion
Ref Expression
elsb4 ([x / y]z yz x)
Distinct variable group:   y,z

Proof of Theorem elsb4
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 nfv 1619 . . 3 y z w
21sbco2 2086 . 2 ([x / y][y / w]z w ↔ [x / w]z w)
3 nfv 1619 . . . 4 w z y
4 elequ2 1715 . . . 4 (w = y → (z wz y))
53, 4sbie 2038 . . 3 ([y / w]z wz y)
65sbbii 1653 . 2 ([x / y][y / w]z w ↔ [x / y]z y)
7 nfv 1619 . . 3 w z x
8 elequ2 1715 . . 3 (w = x → (z wz x))
97, 8sbie 2038 . 2 ([x / w]z wz x)
102, 6, 93bitr3i 266 1 ([x / y]z yz x)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176  [wsb 1648 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-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 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 This theorem is referenced by: (None)
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