Theorem cbval2 1793

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 22-Dec-2003.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 22-Apr-2018.)

Hypotheses

Ref	Expression
cbval2.1	|- F/zph
cbval2.2	|- F/wph
cbval2.3	|- F/xps
cbval2.4	|- F/yps
cbval2.5	|- ((x = z /\ y = w) -> (ph <-> ps))

Assertion

Ref	Expression
cbval2	|- (A.xA.yph <-> A.zA.wps)

Distinct variable groups:   x,y   y,z   x,w   z,w

Allowed substitution hints:   ph(x,y,z,w)   ps(x,y,z,w)

Proof of Theorem cbval2

Step	Hyp	Ref	Expression
1		cbval2.1	. . 3 |- F/zph
2	1	nfal 1465	. 2 |- F/zA.yph
3		cbval2.3	. . 3 |- F/xps
4	3	nfal 1465	. 2 |- F/xA.wps
5		nfv 1418	. . . . . 6 |- F/w x = z
6		cbval2.2	. . . . . 6 |- F/wph
7	5, 6	nfim 1461	. . . . 5 |- F/w(x = z -> ph)
8		nfv 1418	. . . . . 6 |- F/y x = z
9		cbval2.4	. . . . . 6 |- F/yps
10	8, 9	nfim 1461	. . . . 5 |- F/y(x = z -> ps)
11		cbval2.5	. . . . . . 7 |- ((x = z /\ y = w) -> (ph <-> ps))
12	11	expcom 109	. . . . . 6 |- (y = w -> (x = z -> (ph <-> ps)))
13	12	pm5.74d 171	. . . . 5 |- (y = w -> ((x = z -> ph) <-> (x = z -> ps)))
14	7, 10, 13	cbval 1634	. . . 4 |- (A.y(x = z -> ph) <-> A.w(x = z -> ps))
15		19.21v 1750	. . . 4 |- (A.y(x = z -> ph) <-> (x = z -> A.yph))
16		19.21v 1750	. . . 4 |- (A.w(x = z -> ps) <-> (x = z -> A.wps))
17	14, 15, 16	3bitr3i 199	. . 3 |- ((x = z -> A.yph) <-> (x = z -> A.wps))
18	17	pm5.74ri 170	. 2 |- (x = z -> (A.yph <-> A.wps))
19	2, 4, 18	cbval 1634	1 |- (A.xA.yph <-> A.zA.wps)

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98  A.wal 1240   F/wnf 1346

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425

This theorem depends on definitions:  df-bi 110  df-nf 1347

This theorem is referenced by:  cbval2v  1795