Theorem dfsb7a 2004

Description: An alternate definition of proper substitution df-sb 1773. Similar to dfsb7 2001 in that it involves a dummy variable z, but expressed in terms of A. rather than E.. For a version which only requires F/ z ph rather than z and ph being distinct, see sb7af 2003. (Contributed by Jim Kingdon, 5-Feb-2018.)

Assertion

Ref	Expression
dfsb7a	|- ( [ y / x ] ph <-> A. z ( z = y -> A. x ( x = z -> ph ) ) )

Distinct variable groups:   x, z   y, z   ph, z

Allowed substitution hints:   ph( x, y)

Proof of Theorem dfsb7a

Step	Hyp	Ref	Expression
1		nfv 1538	. 2 |- F/ z ph
2	1	sb7af 2003	1 |- ( [ y / x ] ph <-> A. z ( z = y -> A. x ( x = z -> ph ) ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1361   [wsb 1772

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545

This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773

This theorem is referenced by: (None)