Theorem sb7af 1869

Description: An alternative definition of proper substitution df-sb 1646. Similar to dfsb7a 1870 but does not require that ph and z be distinct. Similar to sb7f 1868 in that it involves a dummy variable z, but expressed in terms of A. rather than E.. (Contributed by Jim Kingdon, 5-Feb-2018.)

Hypothesis

Ref	Expression
sb7af.1	|- F/ z ph

Assertion

Ref	Expression
sb7af	|- ( [ y / x ] ph <-> A. z ( z = y -> A. x ( x = z -> ph ) ) )

Distinct variable groups:   x, z   y, z

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

Proof of Theorem sb7af

Step	Hyp	Ref	Expression
1		sb6 1766	. . 3 |- ( [ z / x ] ph <-> A. x ( x = z -> ph ) )
2	1	sbbii 1648	. 2 |- ( [ y / z ] [ z / x ] ph <-> [ y / z ] A. x ( x = z -> ph ) )
3		sb7af.1	. . 3 |- F/ z ph
4	3	sbco2 1839	. 2 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )
5		sb6 1766	. 2 |- ( [ y / z ] A. x ( x = z -> ph ) <-> A. z ( z = y -> A. x ( x = z -> ph ) ) )
6	2, 4, 5	3bitr3i 199	1 |- ( [ y / x ] ph <-> A. z ( z = y -> A. x ( x = z -> ph ) ) )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   F/wnf 1349   [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-bndl 1399  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:  dfsb7a  1870