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Theorem dfsb7 2119
 Description: An alternate definition of proper substitution df-sb 1649. By introducing a dummy variable z in the definiens, we are able to eliminate any distinct variable restrictions among the variables x, y, and φ of the definiendum. No distinct variable conflicts arise because z effectively insulates x from y. To achieve this, we use a chain of two substitutions in the form of sb5 2100, first z for x then y for z. Compare Definition 2.1'' of [Quine] p. 17, which is obtained from this theorem by applying df-clab 2340. Theorem sb7h 2121 provides a version where φ and z don't have to be distinct. (Contributed by NM, 28-Jan-2004.)
Assertion
Ref Expression
dfsb7 ([y / x]φz(z = y x(x = z φ)))
Distinct variable groups:   x,z   y,z   φ,z
Allowed substitution hints:   φ(x,y)

Proof of Theorem dfsb7
StepHypRef Expression
1 sb5 2100 . . 3 ([z / x]φx(x = z φ))
21sbbii 1653 . 2 ([y / z][z / x]φ ↔ [y / z]x(x = z φ))
3 nfv 1619 . . 3 zφ
43sbco2 2086 . 2 ([y / z][z / x]φ ↔ [y / x]φ)
5 sb5 2100 . 2 ([y / z]x(x = z φ) ↔ z(z = y x(x = z φ)))
62, 4, 53bitr3i 266 1 ([y / x]φz(z = y x(x = z φ)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541  [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-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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