Metamath Proof Explorer


Theorem pm2.61danel

Description: Deduction eliminating an elementhood in an antecedent. (Contributed by AV, 5-Dec-2021)

Ref Expression
Hypotheses pm2.61danel.1
|- ( ( ph /\ A e. B ) -> ps )
pm2.61danel.2
|- ( ( ph /\ A e/ B ) -> ps )
Assertion pm2.61danel
|- ( ph -> ps )

Proof

Step Hyp Ref Expression
1 pm2.61danel.1
 |-  ( ( ph /\ A e. B ) -> ps )
2 pm2.61danel.2
 |-  ( ( ph /\ A e/ B ) -> ps )
3 df-nel
 |-  ( A e/ B <-> -. A e. B )
4 3 2 sylan2br
 |-  ( ( ph /\ -. A e. B ) -> ps )
5 1 4 pm2.61dan
 |-  ( ph -> ps )