Theorem nfsb4or 1948

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 1871	. 2 |- F/ z [ w / x ] ph
3		sbequ 1769	. 2 |- ( w = y -> ( [ w / x ] ph <-> [ y / x ] ph ) )
4	2, 3	dvelimor 1943	1 |- ( A. z z = y \/ F/ z [ y / x ] ph )

Syntax hints:   \/ wo 665   A.wal 1288   F/wnf 1395   [wsb 1693

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694

This theorem is referenced by: (None)