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Theorem elsb1 2103
Description: Substitution for the first argument of the non-logical predicate in an atomic formula. See elsb2 2104 for substitution for the second argument. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
Assertion
Ref Expression
elsb1 ([y / x]x zy z)
Distinct variable group:   x,z

Proof of Theorem elsb1
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 nfv 1619 . . 3 x w z
21sbco2 2086 . 2 ([y / x][x / w]w z ↔ [y / w]w z)
3 nfv 1619 . . . 4 w x z
4 elequ1 1713 . . . 4 (w = x → (w zx z))
53, 4sbie 2038 . . 3 ([x / w]w zx z)
65sbbii 1653 . 2 ([y / x][x / w]w z ↔ [y / x]x z)
7 nfv 1619 . . 3 w y z
8 elequ1 1713 . . 3 (w = y → (w zy z))
97, 8sbie 2038 . 2 ([y / w]w zy z)
102, 6, 93bitr3i 266 1 ([y / x]x zy z)
Colors of variables: wff setvar class
Syntax hints:  wb 176  [wsb 1648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  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:  cvjust  2348
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