Theorem hbsb4 1937

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 1872	. 2 |- ( [ w / x ] ph -> A. z [ w / x ] ph )
3		sbequ 1769	. 2 |- ( w = y -> ( [ w / x ] ph <-> [ y / x ] ph ) )
4	2, 3	dvelimALT 1935	1 |- ( -. A. z z = y -> ( [ y / x ] ph -> A. z [ y / x ] ph ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1288   [wsb 1693

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694

This theorem is referenced by:  hbsb4t  1938  dvelimf  1940