Theorem hbsb4 2012

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 1949	. 2 |- ( [ w / x ] ph -> A. z [ w / x ] ph )
3		sbequ 1840	. 2 |- ( w = y -> ( [ w / x ] ph <-> [ y / x ] ph ) )
4	2, 3	dvelimALT 2010	1 |- ( -. A. z z = y -> ( [ y / x ] ph -> A. z [ y / x ] ph ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1351   [wsb 1762

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 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763

This theorem is referenced by:  hbsb4t  2013  dvelimf  2015