Theorem cbvexd 1843

Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 1934. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof rewritten by Jim Kingdon, 10-Jun-2018.)

Hypotheses

Ref	Expression
cbvexd.1	|- F/ y ph
cbvexd.2	|- ( ph -> F/ y ps )
cbvexd.3	|- ( ph -> ( x = y -> ( ps <-> ch ) ) )

Assertion

Ref	Expression
cbvexd	|- ( ph -> ( E. x ps <-> E. y ch ) )

Distinct variable groups:   ph, x   ch, x

Allowed substitution hints:   ph( y)   ps( x, y)   ch( y)

Proof of Theorem cbvexd

Step	Hyp	Ref	Expression
1		cbvexd.1	. . 3 |- F/ y ph
2	1	nfri 1452	. 2 |- ( ph -> A. y ph )
3		cbvexd.2	. . 3 |- ( ph -> F/ y ps )
4	3	nfrd 1453	. 2 |- ( ph -> ( ps -> A. y ps ) )
5		cbvexd.3	. 2 |- ( ph -> ( x = y -> ( ps <-> ch ) ) )
6	2, 4, 5	cbvexdh 1842	1 |- ( ph -> ( E. x ps <-> E. y ch ) )

Syntax hints:   -> wi 4   <-> wb 103   F/wnf 1389   E.wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-nf 1390

This theorem is referenced by:  cbvexdva  1845  vtoclgft  2649  bdsepnft  10678  strcollnft  10779