Theorem sbralie 2590

Description: Implicit to explicit substitution that swaps variables in a quantified expression. (Contributed by NM, 5-Sep-2004.)

Hypothesis

Ref	Expression
sbralie.1	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
sbralie	|- ( [ x / y ] A. x e. y ph <-> A. y e. x ps )

Distinct variable groups:   x, y   ph, y   ps, x

Allowed substitution hints:   ph( x)   ps( y)

Proof of Theorem sbralie

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvralsv 2588	. . . 4 |- ( A. x e. y ph <-> A. z e. y [ z / x ] ph )
2	1	sbbii 1688	. . 3 |- ( [ x / y ] A. x e. y ph <-> [ x / y ] A. z e. y [ z / x ] ph )
3		nfv 1461	. . . 4 |- F/ y A. z e. x [ z / x ] ph
4		raleq 2549	. . . 4 |- ( y = x -> ( A. z e. y [ z / x ] ph <-> A. z e. x [ z / x ] ph ) )
5	3, 4	sbie 1714	. . 3 |- ( [ x / y ] A. z e. y [ z / x ] ph <-> A. z e. x [ z / x ] ph )
6	2, 5	bitri 182	. 2 |- ( [ x / y ] A. x e. y ph <-> A. z e. x [ z / x ] ph )
7		cbvralsv 2588	. . 3 |- ( A. z e. x [ z / x ] ph <-> A. y e. x [ y / z ] [ z / x ] ph )
8		nfv 1461	. . . . . 6 |- F/ z ph
9	8	sbco2 1880	. . . . 5 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )
10		nfv 1461	. . . . . 6 |- F/ x ps
11		sbralie.1	. . . . . 6 |- ( x = y -> ( ph <-> ps ) )
12	10, 11	sbie 1714	. . . . 5 |- ( [ y / x ] ph <-> ps )
13	9, 12	bitri 182	. . . 4 |- ( [ y / z ] [ z / x ] ph <-> ps )
14	13	ralbii 2372	. . 3 |- ( A. y e. x [ y / z ] [ z / x ] ph <-> A. y e. x ps )
15	7, 14	bitri 182	. 2 |- ( A. z e. x [ z / x ] ph <-> A. y e. x ps )
16	6, 15	bitri 182	1 |- ( [ x / y ] A. x e. y ph <-> A. y e. x ps )

Syntax hints:   -> wi 4   <-> wb 103   [wsb 1685   A.wral 2348

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-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  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353

This theorem is referenced by: (None)