Theorem sbralie 2710

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

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

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 2708	. . 3 |- ( A. y e. x ps <-> A. z e. x [ z / y ] ps )
2	1	sbbii 1753	. 2 |- ( [ y / x ] A. y e. x ps <-> [ y / x ] A. z e. x [ z / y ] ps )
3		nfv 1516	. . 3 |- F/ x A. z e. y [ z / y ] ps
4		raleq 2661	. . 3 |- ( x = y -> ( A. z e. x [ z / y ] ps <-> A. z e. y [ z / y ] ps ) )
5	3, 4	sbie 1779	. 2 |- ( [ y / x ] A. z e. x [ z / y ] ps <-> A. z e. y [ z / y ] ps )
6		cbvralsv 2708	. . 3 |- ( A. z e. y [ z / y ] ps <-> A. x e. y [ x / z ] [ z / y ] ps )
7		nfv 1516	. . . . . 6 |- F/ z ps
8	7	sbco2 1953	. . . . 5 |- ( [ x / z ] [ z / y ] ps <-> [ x / y ] ps )
9		nfv 1516	. . . . . 6 |- F/ y ph
10		sbralie.1	. . . . . . . 8 |- ( x = y -> ( ph <-> ps ) )
11	10	bicomd 140	. . . . . . 7 |- ( x = y -> ( ps <-> ph ) )
12	11	equcoms 1696	. . . . . 6 |- ( y = x -> ( ps <-> ph ) )
13	9, 12	sbie 1779	. . . . 5 |- ( [ x / y ] ps <-> ph )
14	8, 13	bitri 183	. . . 4 |- ( [ x / z ] [ z / y ] ps <-> ph )
15	14	ralbii 2472	. . 3 |- ( A. x e. y [ x / z ] [ z / y ] ps <-> A. x e. y ph )
16	6, 15	bitri 183	. 2 |- ( A. z e. y [ z / y ] ps <-> A. x e. y ph )
17	2, 5, 16	3bitrri 206	1 |- ( A. x e. y ph <-> [ y / x ] A. y e. x ps )

Syntax hints:   -> wi 4   <-> wb 104   [wsb 1750   A.wral 2444

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-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  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449

This theorem is referenced by: (None)