Theorem cbval2 2279

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/ z ph
cbval2.2	|- F/ w ph
cbval2.3	|- F/ x ps
cbval2.4	|- F/ y ps
cbval2.5	|- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbval2	|- ( A. x A. y ph <-> A. z A. w ps )

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/ z ph
2	1	nfal 2153	. 2 |- F/ z A. y ph
3		cbval2.3	. . 3 |- F/ x ps
4	3	nfal 2153	. 2 |- F/ x A. w ps
5		nfv 1843	. . . . . 6 |- F/ w x = z
6		cbval2.2	. . . . . 6 |- F/ w ph
7	5, 6	nfim 1825	. . . . 5 |- F/ w ( x = z -> ph )
8		nfv 1843	. . . . . 6 |- F/ y x = z
9		cbval2.4	. . . . . 6 |- F/ y ps
10	8, 9	nfim 1825	. . . . 5 |- F/ y ( x = z -> ps )
11		cbval2.5	. . . . . . 7 |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )
12	11	expcom 451	. . . . . 6 |- ( y = w -> ( x = z -> ( ph <-> ps ) ) )
13	12	pm5.74d 262	. . . . 5 |- ( y = w -> ( ( x = z -> ph ) <-> ( x = z -> ps ) ) )
14	7, 10, 13	cbval 2271	. . . 4 |- ( A. y ( x = z -> ph ) <-> A. w ( x = z -> ps ) )
15		19.21v 1868	. . . 4 |- ( A. y ( x = z -> ph ) <-> ( x = z -> A. y ph ) )
16		19.21v 1868	. . . 4 |- ( A. w ( x = z -> ps ) <-> ( x = z -> A. w ps ) )
17	14, 15, 16	3bitr3i 290	. . 3 |- ( ( x = z -> A. y ph ) <-> ( x = z -> A. w ps ) )
18	17	pm5.74ri 261	. 2 |- ( x = z -> ( A. y ph <-> A. w ps ) )
19	2, 4, 18	cbval 2271	1 |- ( A. x A. y ph <-> A. z A. w ps )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   F/wnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by:  cbvex2  2280  cbval2vOLD  2286  eqrelf  34020