Theorem dfsb7a 1967

Description: An alternate definition of proper substitution df-sb 1736. Similar to dfsb7 1964 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 1966. (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 1508	. 2 |- F/ z ph
2	1	sb7af 1966	1 |- ( [ y / x ] ph <-> A. z ( z = y -> A. x ( x = z -> ph ) ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1329   [wsb 1735

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736

This theorem is referenced by: (None)