Theorem sbralie 3184

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 3182	. . . 4 |- ( A. x e. y ph <-> A. z e. y [ z / x ] ph )
2	1	sbbii 1887	. . 3 |- ( [ x / y ] A. x e. y ph <-> [ x / y ] A. z e. y [ z / x ] ph )
3		nfv 1843	. . . 4 |- F/ y A. z e. x [ z / x ] ph
4		raleq 3138	. . . 4 |- ( y = x -> ( A. z e. y [ z / x ] ph <-> A. z e. x [ z / x ] ph ) )
5	3, 4	sbie 2408	. . 3 |- ( [ x / y ] A. z e. y [ z / x ] ph <-> A. z e. x [ z / x ] ph )
6	2, 5	bitri 264	. 2 |- ( [ x / y ] A. x e. y ph <-> A. z e. x [ z / x ] ph )
7		cbvralsv 3182	. . 3 |- ( A. z e. x [ z / x ] ph <-> A. y e. x [ y / z ] [ z / x ] ph )
8		nfv 1843	. . . . . 6 |- F/ z ph
9	8	sbco2 2415	. . . . 5 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )
10		nfv 1843	. . . . . 6 |- F/ x ps
11		sbralie.1	. . . . . 6 |- ( x = y -> ( ph <-> ps ) )
12	10, 11	sbie 2408	. . . . 5 |- ( [ y / x ] ph <-> ps )
13	9, 12	bitri 264	. . . 4 |- ( [ y / z ] [ z / x ] ph <-> ps )
14	13	ralbii 2980	. . 3 |- ( A. y e. x [ y / z ] [ z / x ] ph <-> A. y e. x ps )
15	7, 14	bitri 264	. 2 |- ( A. z e. x [ z / x ] ph <-> A. y e. x ps )
16	6, 15	bitri 264	1 |- ( [ x / y ] A. x e. y ph <-> A. y e. x ps )

Syntax hints:   -> wi 4   <-> wb 196   [wsb 1880   A.wral 2912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917

This theorem is referenced by:  tfinds2  7063