Theorem cbvexd 1802

Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 1893. (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 1412	. 2 |- ( ph -> A. y ph )
3		cbvexd.2	. . 3 |- ( ph -> F/ y ps )
4	3	nfrd 1413	. 2 |- ( ph -> ( ps -> A. y ps ) )
5		cbvexd.3	. 2 |- ( ph -> ( x = y -> ( ps <-> ch ) ) )
6	2, 4, 5	cbvexdh 1801	1 |- ( ph -> ( E. x ps <-> E. y ch ) )

Syntax hints:   -> wi 4   <-> wb 98   F/wnf 1349   E.wex 1381

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 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-nf 1350

This theorem is referenced by:  cbvexdva  1804  vtoclgft  2604  bdsepnft  10007  strcollnft  10109