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 xφ
dvelimdc.2 zφ
dvelimdc.3 φ_xA
dvelimdc.4 φ_zB
dvelimdc.5 φz=yA=B
Assertion dvelimdc φ¬xx=y_xB

Proof

Step Hyp Ref Expression
1 dvelimdc.1 xφ
2 dvelimdc.2 zφ
3 dvelimdc.3 φ_xA
4 dvelimdc.4 φ_zB
5 dvelimdc.5 φz=yA=B
6 nfv wφ¬xx=y
7 3 nfcrd φxwA
8 4 nfcrd φzwB
9 eleq2 A=BwAwB
10 5 9 syl6 φz=ywAwB
11 1 2 7 8 10 dvelimdf φ¬xx=yxwB
12 11 imp φ¬xx=yxwB
13 6 12 nfcd φ¬xx=y_xB
14 13 ex φ¬xx=y_xB