Theorem hbsb4 2040

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 1977	. 2 |- ( [ w / x ] ph -> A. z [ w / x ] ph )
3		sbequ 1863	. 2 |- ( w = y -> ( [ w / x ] ph <-> [ y / x ] ph ) )
4	2, 3	dvelimALT 2038	1 |- ( -. A. z z = y -> ( [ y / x ] ph -> A. z [ y / x ] ph ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1371   [wsb 1785

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 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786

This theorem is referenced by:  hbsb4t  2041  dvelimf  2043