Theorem sbralie 1439

Description: Implicit to explicit substitution that swaps variables in a quantified expression.

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 group(s):   x,y   ph,y   ps,x

Proof of Theorem sbralie

Step	Hyp	Ref	Expression
1		ax-17 925	. . . . 5 |- (ph -> A.zph)
2		hbs1 986	. . . . 5 |- ([z / x]ph -> A.x[z / x]ph)
3		sbequ12 865	. . . . 5 |- (x = z -> (ph <-> [z / x]ph))
4	1, 2, 3	cbvral 1331	. . . 4 |- (A.x e. y ph <-> A.z e. y [z / x]ph)
5	4	bisb 855	. . 3 |- ([x / y]A.x e. y ph <-> [x / y]A.z e. y [z / x]ph)
6		ax-17 925	. . . 4 |- (A.z e. x [z / x]ph -> A.yA.z e. x [z / x]ph)
7		raleq 1324	. . . 4 |- (y = x -> (A.z e. y [z / x]ph <-> A.z e. x [z / x]ph))
8	6, 7	sbie 904	. . 3 |- ([x / y]A.z e. y [z / x]ph <-> A.z e. x [z / x]ph)
9	5, 8	bitr 151	. 2 |- ([x / y]A.x e. y ph <-> A.z e. x [z / x]ph)
10		ax-17 925	. . 3 |- ([z / x]ph -> A.y[z / x]ph)
11		hbs1 986	. . 3 |- ([y / z][z / x]ph -> A.z[y / z][z / x]ph)
12		sbequ12 865	. . 3 |- (z = y -> ([z / x]ph <-> [y / z][z / x]ph))
13	10, 11, 12	cbvral 1331	. 2 |- (A.z e. x [z / x]ph <-> A.y e. x [y / z][z / x]ph)
14	1	sbco2 913	. . . 4 |- ([y / z][z / x]ph <-> [y / x]ph)
15		ax-17 925	. . . . 5 |- (ps -> A.xps)
16		sbralie.1	. . . . 5 |- (x = y -> (ph <-> ps))
17	15, 16	sbie 904	. . . 4 |- ([y / x]ph <-> ps)
18	14, 17	bitr 151	. . 3 |- ([y / z][z / x]ph <-> ps)
19	18	biral 1223	. 2 |- (A.y e. x [y / z][z / x]ph <-> A.y e. x ps)
20	9, 13, 19	3bitr 155	1 |- ([x / y]A.x e. y ph <-> A.y e. x ps)

Syntax hints:   -> wi 2   <-> wb 127   = weq 797  [wsb 852  A.wral 1201

This theorem is referenced by:  tfinds2 2405

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-cleq 1097  df-clel 1099  df-ral 1205

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