Theorem hbsb4 1988

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 1923	. 2 |- ( [ w / x ] ph -> A. z [ w / x ] ph )
3		sbequ 1813	. 2 |- ( w = y -> ( [ w / x ] ph <-> [ y / x ] ph ) )
4	2, 3	dvelimALT 1986	1 |- ( -. A. z z = y -> ( [ y / x ] ph -> A. z [ y / x ] ph ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1330   [wsb 1736

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-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737

This theorem is referenced by:  hbsb4t  1989  dvelimf  1991