Theorem sbal1 1878

Description: A theorem used in elimination of disjoint variable restriction on x and 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 1876	. . . 4 |- ( [ w / y ] A. x ph <-> A. x [ w / y ] ph )
2	1	sbbii 1648	. . 3 |- ( [ z / w ] [ w / y ] A. x ph <-> [ z / w ] A. x [ w / y ] ph )
3		sbal1yz 1877	. . 3 |- ( -. A. x x = z -> ( [ z / w ] A. x [ w / y ] ph <-> A. x [ z / w ] [ w / y ] ph ) )
4	2, 3	syl5bb 181	. 2 |- ( -. A. x x = z -> ( [ z / w ] [ w / y ] A. x ph <-> A. x [ z / w ] [ w / y ] ph ) )
5		ax-17 1419	. . 3 |- ( A. x ph -> A. w A. x ph )
6	5	sbco2v 1821	. 2 |- ( [ z / w ] [ w / y ] A. x ph <-> [ z / y ] A. x ph )
7		ax-17 1419	. . . 4 |- ( ph -> A. w ph )
8	7	sbco2v 1821	. . 3 |- ( [ z / w ] [ w / y ] ph <-> [ z / y ] ph )
9	8	albii 1359	. 2 |- ( A. x [ z / w ] [ w / y ] ph <-> A. x [ z / y ] ph )
10	4, 6, 9	3bitr3g 211	1 |- ( -. A. x x = z -> ( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 98   A.wal 1241   [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646

This theorem is referenced by: (None)