Theorem dfsb7 2455

Description: An alternate definition of proper substitution df-sb 1881. By introducing a dummy variable z in the definiens, we are able to eliminate any distinct variable restrictions among the variables x, y, and ph 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 2430, 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 2609. Theorem sb7h 2454 provides a version where ph and z don't have to be distinct. (Contributed by NM, 28-Jan-2004.)

Assertion

Ref	Expression
dfsb7	|- ( [ y / x ] ph <-> E. z ( z = y /\ E. x ( x = z /\ ph ) ) )

Distinct variable groups:   y, z   ph, z

Allowed substitution hints:   ph( x, y)

Proof of Theorem dfsb7

Step	Hyp	Ref	Expression
1		nfv 1843	. 2 |- F/ z ph
2	1	sb7f 2453	1 |- ( [ y / x ] ph <-> E. z ( z = y /\ E. x ( x = z /\ ph ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   E.wex 1704   [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by: (None)