Metamath Proof Explorer


Theorem dvelimalcased

Description: Eliminate a disjoint variable condition from a universally quantified statement using cases. (Contributed by BTernaryTau, 31-Jul-2025)

Ref Expression
Hypotheses dvelimalcased.1
|- F/ x ph
dvelimalcased.2
|- ( -. A. x x = y -> F/ z ph )
dvelimalcased.3
|- ( ( ph /\ -. A. x x = y ) -> F/ x ps )
dvelimalcased.4
|- ( ( ph /\ -. A. x x = y ) -> F/ z th )
dvelimalcased.5
|- ( ( ph /\ -. A. x x = y ) -> ( z = x -> ( ps -> th ) ) )
dvelimalcased.6
|- ( ( ph /\ A. x x = y ) -> ( ch -> th ) )
dvelimalcased.7
|- ( ph -> A. z ps )
dvelimalcased.8
|- ( ph -> A. x ch )
Assertion dvelimalcased
|- ( ph -> A. x th )

Proof

Step Hyp Ref Expression
1 dvelimalcased.1
 |-  F/ x ph
2 dvelimalcased.2
 |-  ( -. A. x x = y -> F/ z ph )
3 dvelimalcased.3
 |-  ( ( ph /\ -. A. x x = y ) -> F/ x ps )
4 dvelimalcased.4
 |-  ( ( ph /\ -. A. x x = y ) -> F/ z th )
5 dvelimalcased.5
 |-  ( ( ph /\ -. A. x x = y ) -> ( z = x -> ( ps -> th ) ) )
6 dvelimalcased.6
 |-  ( ( ph /\ A. x x = y ) -> ( ch -> th ) )
7 dvelimalcased.7
 |-  ( ph -> A. z ps )
8 dvelimalcased.8
 |-  ( ph -> A. x ch )
9 nfa1
 |-  F/ x A. x x = y
10 1 9 nfan
 |-  F/ x ( ph /\ A. x x = y )
11 10 6 alimd
 |-  ( ( ph /\ A. x x = y ) -> ( A. x ch -> A. x th ) )
12 11 ex
 |-  ( ph -> ( A. x x = y -> ( A. x ch -> A. x th ) ) )
13 8 12 mpid
 |-  ( ph -> ( A. x x = y -> A. x th ) )
14 nfv
 |-  F/ z -. A. x x = y
15 14 2 nfan1c
 |-  F/ z ( ph /\ -. A. x x = y )
16 nfna1
 |-  F/ x -. A. x x = y
17 1 16 nfan
 |-  F/ x ( ph /\ -. A. x x = y )
18 15 17 3 4 5 cbv1v
 |-  ( ( ph /\ -. A. x x = y ) -> ( A. z ps -> A. x th ) )
19 18 ex
 |-  ( ph -> ( -. A. x x = y -> ( A. z ps -> A. x th ) ) )
20 7 19 mpid
 |-  ( ph -> ( -. A. x x = y -> A. x th ) )
21 13 20 pm2.61d
 |-  ( ph -> A. x th )