Theorem sbal1 1990

Description: A theorem used in elimination of disjoint variable conditions on x , y by replacing it with a distinctor -. A. x x = z. (Contributed by NM, 5-Aug-1993.) (Proof rewitten by Jim Kingdon, 24-Feb-2018.)

Assertion

Ref	Expression
sbal1	|- ( -. A. x x = z -> ( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y, z)

Proof of Theorem sbal1

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbal 1988	. . . 4 |- ( [ w / y ] A. x ph <-> A. x [ w / y ] ph )
2	1	sbbii 1753	. . 3 |- ( [ z / w ] [ w / y ] A. x ph <-> [ z / w ] A. x [ w / y ] ph )
3		sbal1yz 1989	. . 3 |- ( -. A. x x = z -> ( [ z / w ] A. x [ w / y ] ph <-> A. x [ z / w ] [ w / y ] ph ) )
4	2, 3	syl5bb 191	. 2 |- ( -. A. x x = z -> ( [ z / w ] [ w / y ] A. x ph <-> A. x [ z / w ] [ w / y ] ph ) )
5		ax-17 1514	. . 3 |- ( A. x ph -> A. w A. x ph )
6	5	sbco2vh 1933	. 2 |- ( [ z / w ] [ w / y ] A. x ph <-> [ z / y ] A. x ph )
7		ax-17 1514	. . . 4 |- ( ph -> A. w ph )
8	7	sbco2vh 1933	. . 3 |- ( [ z / w ] [ w / y ] ph <-> [ z / y ] ph )
9	8	albii 1458	. 2 |- ( A. x [ z / w ] [ w / y ] ph <-> A. x [ z / y ] ph )
10	4, 6, 9	3bitr3g 221	1 |- ( -. A. x x = z -> ( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104   A.wal 1341   [wsb 1750

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751

This theorem is referenced by: (None)