Theorem sb7f 1909

Description: This version of dfsb7 1908 does not require that ph and z be distinct. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-17 1459 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1686 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.) (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypothesis

Ref	Expression
sb7f.1	|- ( ph -> A. z ph )

Assertion

Ref	Expression
sb7f	|- ( [ y / x ] ph <-> E. z ( z = y /\ E. x ( x = z /\ ph ) ) )

Distinct variable groups:   x, z   y, z

Allowed substitution hints:   ph( x, y, z)

Proof of Theorem sb7f

Step	Hyp	Ref	Expression
1		sb5 1808	. . 3 |- ( [ z / x ] ph <-> E. x ( x = z /\ ph ) )
2	1	sbbii 1688	. 2 |- ( [ y / z ] [ z / x ] ph <-> [ y / z ] E. x ( x = z /\ ph ) )
3		sb7f.1	. . 3 |- ( ph -> A. z ph )
4	3	sbco2v 1862	. 2 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )
5		sb5 1808	. 2 |- ( [ y / z ] E. x ( x = z /\ ph ) <-> E. z ( z = y /\ E. x ( x = z /\ ph ) ) )
6	2, 4, 5	3bitr3i 208	1 |- ( [ y / x ] ph <-> E. z ( z = y /\ E. x ( x = z /\ ph ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1282   E.wex 1421   [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686

This theorem is referenced by: (None)