Theorem sbralie 1941

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 groups:   x,y   ph,y   ps,x

Proof of Theorem sbralie

Step	Hyp	Ref	Expression
1		ax-17 971	. . . . 5 |- (ph -> A.zph)
2		hbs1 1332	. . . . 5 |- ([z / x]ph -> A.x[z / x]ph)
3		sbequ12 1181	. . . . 5 |- (x = z -> (ph <-> [z / x]ph))
4	1, 2, 3	cbvral 1798	. . . 4 |- (A.x e. y ph <-> A.z e. y [z / x]ph)
5	4	sbbii 1174	. . 3 |- ([x / y]A.x e. y ph <-> [x / y]A.z e. y [z / x]ph)
6		ax-17 971	. . . 4 |- (A.z e. x [z / x]ph -> A.yA.z e. x [z / x]ph)
7		raleq1 1786	. . . 4 |- (y = x -> (A.z e. y [z / x]ph <-> A.z e. x [z / x]ph))
8	6, 7	sbie 1196	. . 3 |- ([x / y]A.z e. y [z / x]ph <-> A.z e. x [z / x]ph)
9	5, 8	bitr 173	. 2 |- ([x / y]A.x e. y ph <-> A.z e. x [z / x]ph)
10		ax-17 971	. . 3 |- ([z / x]ph -> A.y[z / x]ph)
11		hbs1 1332	. . 3 |- ([y / z][z / x]ph -> A.z[y / z][z / x]ph)
12		sbequ12 1181	. . 3 |- (z = y -> ([z / x]ph <-> [y / z][z / x]ph))
13	10, 11, 12	cbvral 1798	. 2 |- (A.z e. x [z / x]ph <-> A.y e. x [y / z][z / x]ph)
14	1	sbco2 1255	. . . 4 |- ([y / z][z / x]ph <-> [y / x]ph)
15		ax-17 971	. . . . 5 |- (ps -> A.xps)
16		sbralie.1	. . . . 5 |- (x = y -> (ph <-> ps))
17	15, 16	sbie 1196	. . . 4 |- ([y / x]ph <-> ps)
18	14, 17	bitr 173	. . 3 |- ([y / z][z / x]ph <-> ps)
19	18	ralbii 1667	. 2 |- (A.y e. x [y / z][z / x]ph <-> A.y e. x ps)
20	9, 13, 19	3bitr 177	1 |- ([x / y]A.x e. y ph <-> A.y e. x ps)

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956  [wsbc 1170  A.wral 1645

This theorem is referenced by:  tfinds2 3165

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-cleq 1469  df-clel 1472  df-ral 1649

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