Theorem sbal1 1978

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 1976	. . . 4 |- ( [ w / y ] A. x ph <-> A. x [ w / y ] ph )
2	1	sbbii 1739	. . 3 |- ( [ z / w ] [ w / y ] A. x ph <-> [ z / w ] A. x [ w / y ] ph )
3		sbal1yz 1977	. . 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 1507	. . 3 |- ( A. x ph -> A. w A. x ph )
6	5	sbco2vh 1919	. 2 |- ( [ z / w ] [ w / y ] A. x ph <-> [ z / y ] A. x ph )
7		ax-17 1507	. . . 4 |- ( ph -> A. w ph )
8	7	sbco2vh 1919	. . 3 |- ( [ z / w ] [ w / y ] ph <-> [ z / y ] ph )
9	8	albii 1447	. 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 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: (None)