Theorem sbralie 2848

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 2846	. . . 4 |- (A.x e. y ph <-> A.z e. y [z / x]ph)
2	1	sbbii 1653	. . 3 |- ([x / y]A.x e. y ph <-> [x / y]A.z e. y [z / x]ph)
3		nfv 1619	. . . 4 |- F/yA.z e. x [z / x]ph
4		raleq 2807	. . . 4 |- (y = x -> (A.z e. y [z / x]ph <-> A.z e. x [z / x]ph))
5	3, 4	sbie 2038	. . 3 |- ([x / y]A.z e. y [z / x]ph <-> A.z e. x [z / x]ph)
6	2, 5	bitri 240	. 2 |- ([x / y]A.x e. y ph <-> A.z e. x [z / x]ph)
7		cbvralsv 2846	. . 3 |- (A.z e. x [z / x]ph <-> A.y e. x [y / z][z / x]ph)
8		nfv 1619	. . . . . 6 |- F/zph
9	8	sbco2 2086	. . . . 5 |- ([y / z][z / x]ph <-> [y / x]ph)
10		nfv 1619	. . . . . 6 |- F/xps
11		sbralie.1	. . . . . 6 |- (x = y -> (ph <-> ps))
12	10, 11	sbie 2038	. . . . 5 |- ([y / x]ph <-> ps)
13	9, 12	bitri 240	. . . 4 |- ([y / z][z / x]ph <-> ps)
14	13	ralbii 2638	. . 3 |- (A.y e. x [y / z][z / x]ph <-> A.y e. x ps)
15	7, 14	bitri 240	. 2 |- (A.z e. x [z / x]ph <-> A.y e. x ps)
16	6, 15	bitri 240	1 |- ([x / y]A.x e. y ph <-> A.y e. x ps)

Syntax hints:   -> wi 4   <-> wb 176  [wsb 1648  A.wral 2614

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619

This theorem is referenced by: (None)