# Metamath Proof Explorer

## Theorem cbvexdw

Description: Deduction used to change bound variables, using implicit substitution. Version of cbvexd with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 2-Jan-2002) (Revised by Gino Giotto, 10-Jan-2024)

Ref Expression
Hypotheses cbvaldw.1
`|- F/ y ph`
cbvaldw.2
`|- ( ph -> F/ y ps )`
cbvaldw.3
`|- ( ph -> ( x = y -> ( ps <-> ch ) ) )`
Assertion cbvexdw
`|- ( ph -> ( E. x ps <-> E. y ch ) )`

### Proof

Step Hyp Ref Expression
1 cbvaldw.1
` |-  F/ y ph`
2 cbvaldw.2
` |-  ( ph -> F/ y ps )`
3 cbvaldw.3
` |-  ( ph -> ( x = y -> ( ps <-> ch ) ) )`
4 2 nfnd
` |-  ( ph -> F/ y -. ps )`
5 notbi
` |-  ( ( ps <-> ch ) <-> ( -. ps <-> -. ch ) )`
6 3 5 syl6ib
` |-  ( ph -> ( x = y -> ( -. ps <-> -. ch ) ) )`
7 1 4 6 cbvaldw
` |-  ( ph -> ( A. x -. ps <-> A. y -. ch ) )`
8 alnex
` |-  ( A. x -. ps <-> -. E. x ps )`
9 alnex
` |-  ( A. y -. ch <-> -. E. y ch )`
10 7 8 9 3bitr3g
` |-  ( ph -> ( -. E. x ps <-> -. E. y ch ) )`
11 10 con4bid
` |-  ( ph -> ( E. x ps <-> E. y ch ) )`