Theorem nfsb4or 2009

Description: A variable not free remains so after substitution with a distinct variable. (Contributed by Jim Kingdon, 11-May-2018.)

Hypothesis

Ref	Expression
nfsb4or.1	|- F/ z ph

Assertion

Ref	Expression
nfsb4or	|- ( A. z z = y \/ F/ z [ y / x ] ph )

Proof of Theorem nfsb4or

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfsb4or.1	. . 3 |- F/ z ph
2	1	nfsb 1934	. 2 |- F/ z [ w / x ] ph
3		sbequ 1828	. 2 |- ( w = y -> ( [ w / x ] ph <-> [ y / x ] ph ) )
4	2, 3	dvelimor 2006	1 |- ( A. z z = y \/ F/ z [ y / x ] ph )

Syntax hints:   \/ wo 698   A.wal 1341   F/wnf 1448   [wsb 1750

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751

This theorem is referenced by: (None)