Theorem dfsb7 1867

Description: An alternate definition of proper substitution df-sb 1646. 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 1767, first z for x then y for z. Compare Definition 2.1'' of [Quine] p. 17. Theorem sb7f 1868 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:   x, z   y, z   ph, z

Allowed substitution hints:   ph( x, y)

Proof of Theorem dfsb7

Step	Hyp	Ref	Expression
1		sb5 1767	. . 3 |- ( [ z / x ] ph <-> E. x ( x = z /\ ph ) )
2	1	sbbii 1648	. 2 |- ( [ y / z ] [ z / x ] ph <-> [ y / z ] E. x ( x = z /\ ph ) )
3		ax-17 1419	. . 3 |- ( ph -> A. z ph )
4	3	sbco2v 1821	. 2 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )
5		sb5 1767	. 2 |- ( [ y / z ] E. x ( x = z /\ ph ) <-> E. z ( z = y /\ E. x ( x = z /\ ph ) ) )
6	2, 4, 5	3bitr3i 199	1 |- ( [ y / x ] ph <-> E. z ( z = y /\ E. x ( x = z /\ ph ) ) )

Syntax hints:   /\ wa 97   <-> wb 98   E.wex 1381   [wsb 1645

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646

This theorem is referenced by: (None)