Theorem cbvexd 1920

Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2010. (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 1512	. 2 |- ( ph -> A. y ph )
3		cbvexd.2	. . 3 |- ( ph -> F/ y ps )
4	3	nfrd 1513	. 2 |- ( ph -> ( ps -> A. y ps ) )
5		cbvexd.3	. 2 |- ( ph -> ( x = y -> ( ps <-> ch ) ) )
6	2, 4, 5	cbvexdh 1919	1 |- ( ph -> ( E. x ps <-> E. y ch ) )

Syntax hints:   -> wi 4   <-> wb 104   F/wnf 1453   E.wex 1485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-nf 1454

This theorem is referenced by:  cbvexdva  1922  vtoclgft  2780  bdsepnft  13922  strcollnft  14019