Theorem cbvex2 1947

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 6-Oct-2016.)

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
cbvex2	|- ( E. x E. y ph <-> E. z E. 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 cbvex2

Step	Hyp	Ref	Expression
1		cbval2.1	. . 3 |- F/ z ph
2	1	nfex 1661	. 2 |- F/ z E. y ph
3		cbval2.3	. . 3 |- F/ x ps
4	3	nfex 1661	. 2 |- F/ x E. w ps
5		nfv 1552	. . . . . 6 |- F/ w x = z
6		cbval2.2	. . . . . 6 |- F/ w ph
7	5, 6	nfan 1589	. . . . 5 |- F/ w ( x = z /\ ph )
8		nfv 1552	. . . . . 6 |- F/ y x = z
9		cbval2.4	. . . . . 6 |- F/ y ps
10	8, 9	nfan 1589	. . . . 5 |- F/ y ( x = z /\ ps )
11		cbval2.5	. . . . . . 7 |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )
12	11	expcom 116	. . . . . 6 |- ( y = w -> ( x = z -> ( ph <-> ps ) ) )
13	12	pm5.32d 450	. . . . 5 |- ( y = w -> ( ( x = z /\ ph ) <-> ( x = z /\ ps ) ) )
14	7, 10, 13	cbvex 1780	. . . 4 |- ( E. y ( x = z /\ ph ) <-> E. w ( x = z /\ ps ) )
15		19.42v 1931	. . . 4 |- ( E. y ( x = z /\ ph ) <-> ( x = z /\ E. y ph ) )
16		19.42v 1931	. . . 4 |- ( E. w ( x = z /\ ps ) <-> ( x = z /\ E. w ps ) )
17	14, 15, 16	3bitr3i 210	. . 3 |- ( ( x = z /\ E. y ph ) <-> ( x = z /\ E. w ps ) )
18		pm5.32 453	. . 3 |- ( ( x = z -> ( E. y ph <-> E. w ps ) ) <-> ( ( x = z /\ E. y ph ) <-> ( x = z /\ E. w ps ) ) )
19	17, 18	mpbir 146	. 2 |- ( x = z -> ( E. y ph <-> E. w ps ) )
20	2, 4, 19	cbvex 1780	1 |- ( E. x E. y ph <-> E. z E. w ps )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   F/wnf 1484   E.wex 1516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558

This theorem depends on definitions:  df-bi 117  df-nf 1485

This theorem is referenced by:  cbvex2v  1949  cbvopab  4120  cbvoprab12  6029