Theorem dfsb7a 1970

Description: An alternate definition of proper substitution df-sb 1737. Similar to dfsb7 1967 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 1969. (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 1509	. 2 |- F/ z ph
2	1	sb7af 1969	1 |- ( [ y / x ] ph <-> A. z ( z = y -> A. x ( x = z -> ph ) ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1330   [wsb 1736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737

This theorem is referenced by: (None)