Theorem nfsb4or 1940

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 1863	. 2 |- F/ z [ w / x ] ph
3		sbequ 1761	. 2 |- ( w = y -> ( [ w / x ] ph <-> [ y / x ] ph ) )
4	2, 3	dvelimor 1935	1 |- ( A. z z = y \/ F/ z [ y / x ] ph )

Syntax hints:   \/ wo 661   A.wal 1282   F/wnf 1389   [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686

This theorem is referenced by: (None)