Metamath Proof Explorer

Theorem rzal

Description: Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997) (Proof shortened by Andrew Salmon, 26-Jun-2011)

Ref Expression
Assertion rzal 
|- ( A = (/) -> A. x e. A ph )

Proof

Step Hyp Ref Expression
1 ne0i 
 |-  ( x e. A -> A =/= (/) )
2 1 necon2bi 
 |-  ( A = (/) -> -. x e. A )
3 2 pm2.21d 
 |-  ( A = (/) -> ( x e. A -> ph ) )
4 3 ralrimiv 
 |-  ( A = (/) -> A. x e. A ph )