Theorem sb7af 2012

Description: An alternate definition of proper substitution df-sb 1777. Similar to dfsb7a 2013 but does not require that ph and z be distinct. Similar to sb7f 2011 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 1901	. . 3 |- ( [ z / x ] ph <-> A. x ( x = z -> ph ) )
2	1	sbbii 1779	. 2 |- ( [ y / z ] [ z / x ] ph <-> [ y / z ] A. x ( x = z -> ph ) )
3		sb7af.1	. . 3 |- F/ z ph
4	3	sbco2 1984	. 2 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )
5		sb6 1901	. 2 |- ( [ y / z ] A. x ( x = z -> ph ) <-> A. z ( z = y -> A. x ( x = z -> ph ) ) )
6	2, 4, 5	3bitr3i 210	1 |- ( [ y / x ] ph <-> A. z ( z = y -> A. x ( x = z -> ph ) ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1362   F/wnf 1474   [wsb 1776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777

This theorem is referenced by:  dfsb7a  2013