Theorem cbviota 4345

Description: Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)

Hypotheses

Ref	Expression
cbviota.1	|- (x = y -> (ph <-> ps))
cbviota.2	|- F/yph
cbviota.3	|- F/xps

Assertion

Ref	Expression
cbviota	|- (iotaxph) = (iotayps)

Proof of Theorem cbviota

Dummy variables z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1619	. . . . . 6 |- F/z(ph <-> x = w)
2		nfs1v 2106	. . . . . . 7 |- F/x[z / x]ph
3		nfv 1619	. . . . . . 7 |- F/x z = w
4	2, 3	nfbi 1834	. . . . . 6 |- F/x([z / x]ph <-> z = w)
5		sbequ12 1919	. . . . . . 7 |- (x = z -> (ph <-> [z / x]ph))
6		equequ1 1684	. . . . . . 7 |- (x = z -> (x = w <-> z = w))
7	5, 6	bibi12d 312	. . . . . 6 |- (x = z -> ((ph <-> x = w) <-> ([z / x]ph <-> z = w)))
8	1, 4, 7	cbval 1984	. . . . 5 |- (A.x(ph <-> x = w) <-> A.z([z / x]ph <-> z = w))
9		cbviota.2	. . . . . . . 8 |- F/yph
10	9	nfsb 2109	. . . . . . 7 |- F/y[z / x]ph
11		nfv 1619	. . . . . . 7 |- F/y z = w
12	10, 11	nfbi 1834	. . . . . 6 |- F/y([z / x]ph <-> z = w)
13		nfv 1619	. . . . . 6 |- F/z(ps <-> y = w)
14		sbequ 2060	. . . . . . . 8 |- (z = y -> ([z / x]ph <-> [y / x]ph))
15		cbviota.3	. . . . . . . . 9 |- F/xps
16		cbviota.1	. . . . . . . . 9 |- (x = y -> (ph <-> ps))
17	15, 16	sbie 2038	. . . . . . . 8 |- ([y / x]ph <-> ps)
18	14, 17	syl6bb 252	. . . . . . 7 |- (z = y -> ([z / x]ph <-> ps))
19		equequ1 1684	. . . . . . 7 |- (z = y -> (z = w <-> y = w))
20	18, 19	bibi12d 312	. . . . . 6 |- (z = y -> (([z / x]ph <-> z = w) <-> (ps <-> y = w)))
21	12, 13, 20	cbval 1984	. . . . 5 |- (A.z([z / x]ph <-> z = w) <-> A.y(ps <-> y = w))
22	8, 21	bitri 240	. . . 4 |- (A.x(ph <-> x = w) <-> A.y(ps <-> y = w))
23	22	abbii 2466	. . 3 |- {w | A.x(ph <-> x = w)} = {w | A.y(ps <-> y = w)}
24	23	unieqi 3902	. 2 |- U.{w | A.x(ph <-> x = w)} = U.{w | A.y(ps <-> y = w)}
25		dfiota2 4341	. 2 |- (iotaxph) = U.{w | A.x(ph <-> x = w)}
26		dfiota2 4341	. 2 |- (iotayps) = U.{w | A.y(ps <-> y = w)}
27	24, 25, 26	3eqtr4i 2383	1 |- (iotaxph) = (iotayps)

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540   F/wnf 1544   = wceq 1642  [wsb 1648  {cab 2339  U.cuni 3892  iotacio 4338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rex 2621  df-sn 3742  df-uni 3893  df-iota 4340

This theorem is referenced by:  cbviotav  4346