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Theorem dfsb7 1864
Description: An alternate definition of proper substitution df-sb 1643. 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 1764, first z for x then y for z. Compare Definition 2.1'' of [Quine] p. 17. Theorem sb7f 1865 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 1764 . . 3 ([z / x]φx(x = z φ))
21sbbii 1645 . 2 ([y / z][z / x]φ ↔ [y / z]x(x = z φ))
3 ax-17 1416 . . 3 (φzφ)
43sbco2v 1818 . 2 ([y / z][z / x]φ ↔ [y / x]φ)
5 sb5 1764 . 2 ([y / z]x(x = z φ) ↔ z(z = y x(x = z φ)))
62, 4, 53bitr3i 199 1 ([y / x]φz(z = y x(x = z φ)))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98  wex 1378  [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-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643
This theorem is referenced by: (None)
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