Theorem hbsb4 2065

Description: A variable not free remains so after substitution with a distinct variable. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.)

Hypothesis

Ref	Expression
hbsb4.1	|- ( ph -> A. z ph )

Assertion

Ref	Expression
hbsb4	|- ( -. A. z z = y -> ( [ y / x ] ph -> A. z [ y / x ] ph ) )

Proof of Theorem hbsb4

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hbsb4.1	. . 3 |- ( ph -> A. z ph )
2	1	hbsb 2002	. 2 |- ( [ w / x ] ph -> A. z [ w / x ] ph )
3		sbequ 1888	. 2 |- ( w = y -> ( [ w / x ] ph <-> [ y / x ] ph ) )
4	2, 3	dvelimALT 2063	1 |- ( -. A. z z = y -> ( [ y / x ] ph -> A. z [ y / x ] ph ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1395   [wsb 1810

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-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811

This theorem is referenced by:  hbsb4t  2066  dvelimf  2068