Theorem nfsb4or 2081

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 2006	. 2 |- F/ z [ w / x ] ph
3		sbequ 1893	. 2 |- ( w = y -> ( [ w / x ] ph <-> [ y / x ] ph ) )
4	2, 3	dvelimor 2078	1 |- ( A. z z = y \/ F/ z [ y / x ] ph )

Syntax hints:   \/ wo 720   A.wal 1400   F/wnf 1513   [wsb 1815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816

This theorem is referenced by: (None)