Theorem sbal1 1919

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 1917	. . . 4 |- ( [ w / y ] A. x ph <-> A. x [ w / y ] ph )
2	1	sbbii 1688	. . 3 |- ( [ z / w ] [ w / y ] A. x ph <-> [ z / w ] A. x [ w / y ] ph )
3		sbal1yz 1918	. . 3 |- ( -. A. x x = z -> ( [ z / w ] A. x [ w / y ] ph <-> A. x [ z / w ] [ w / y ] ph ) )
4	2, 3	syl5bb 190	. 2 |- ( -. A. x x = z -> ( [ z / w ] [ w / y ] A. x ph <-> A. x [ z / w ] [ w / y ] ph ) )
5		ax-17 1459	. . 3 |- ( A. x ph -> A. w A. x ph )
6	5	sbco2v 1862	. 2 |- ( [ z / w ] [ w / y ] A. x ph <-> [ z / y ] A. x ph )
7		ax-17 1459	. . . 4 |- ( ph -> A. w ph )
8	7	sbco2v 1862	. . 3 |- ( [ z / w ] [ w / y ] ph <-> [ z / y ] ph )
9	8	albii 1399	. 2 |- ( A. x [ z / w ] [ w / y ] ph <-> A. x [ z / y ] ph )
10	4, 6, 9	3bitr3g 220	1 |- ( -. A. x x = z -> ( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 103   A.wal 1282   [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686

This theorem is referenced by: (None)