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Metamath Proof Explorer

Theorem dvelimdc

Description: Deduction form of dvelimc . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by Mario Carneiro, 8-Oct-2016) (New usage is discouraged.)

Ref Expression
Hypotheses dvelimdc.1 
|- F/ x ph
dvelimdc.2 
|- F/ z ph
dvelimdc.3 
|- ( ph -> F/_ x A )
dvelimdc.4 
|- ( ph -> F/_ z B )
dvelimdc.5 
|- ( ph -> ( z = y -> A = B ) )
Assertion dvelimdc 
|- ( ph -> ( -. A. x x = y -> F/_ x B ) )

Proof

Step Hyp Ref Expression
1 dvelimdc.1 
 |-  F/ x ph
2 dvelimdc.2 
 |-  F/ z ph
3 dvelimdc.3 
 |-  ( ph -> F/_ x A )
4 dvelimdc.4 
 |-  ( ph -> F/_ z B )
5 dvelimdc.5 
 |-  ( ph -> ( z = y -> A = B ) )
6 nfv 
 |-  F/ w ( ph /\ -. A. x x = y )
7 3 nfcrd 
 |-  ( ph -> F/ x w e. A )
8 4 nfcrd 
 |-  ( ph -> F/ z w e. B )
9 eleq2 
 |-  ( A = B -> ( w e. A <-> w e. B ) )
10 5 9 syl6 
 |-  ( ph -> ( z = y -> ( w e. A <-> w e. B ) ) )
11 1 2 7 8 10 dvelimdf 
 |-  ( ph -> ( -. A. x x = y -> F/ x w e. B ) )
12 11 imp 
 |-  ( ( ph /\ -. A. x x = y ) -> F/ x w e. B )
13 6 12 nfcd 
 |-  ( ( ph /\ -. A. x x = y ) -> F/_ x B )
14 13 ex 
 |-  ( ph -> ( -. A. x x = y -> F/_ x B ) )