Metamath Proof Explorer


Theorem r19.21v

Description: Restricted quantifier version of 19.21v . (Contributed by NM, 15-Oct-2003) (Proof shortened by Andrew Salmon, 30-May-2011) Reduce dependencies on axioms. (Revised by Wolf Lammen, 2-Jan-2020) (Proof shortened by Wolf Lammen, 11-Dec-2024)

Ref Expression
Assertion r19.21v x A φ ψ φ x A ψ

Proof

Step Hyp Ref Expression
1 pm2.27 φ φ ψ ψ
2 1 ralimdv φ x A φ ψ x A ψ
3 2 com12 x A φ ψ φ x A ψ
4 pm2.21 ¬ φ φ ψ
5 4 ralrimivw ¬ φ x A φ ψ
6 ax-1 ψ φ ψ
7 6 ralimi x A ψ x A φ ψ
8 5 7 ja φ x A ψ x A φ ψ
9 3 8 impbii x A φ ψ φ x A ψ