Metamath Proof Explorer


Theorem nfrexdg

Description: Deduction version of nfrexg . Usage of this theorem is discouraged because it depends on ax-13 . See nfrexd for a version with a disjoint variable condition, but not requiring ax-13 . (Contributed by Mario Carneiro, 14-Oct-2016) (New usage is discouraged.)

Ref Expression
Hypotheses nfrexdg.1 𝑦 𝜑
nfrexdg.2 ( 𝜑 𝑥 𝐴 )
nfrexdg.3 ( 𝜑 → Ⅎ 𝑥 𝜓 )
Assertion nfrexdg ( 𝜑 → Ⅎ 𝑥𝑦𝐴 𝜓 )

Proof

Step Hyp Ref Expression
1 nfrexdg.1 𝑦 𝜑
2 nfrexdg.2 ( 𝜑 𝑥 𝐴 )
3 nfrexdg.3 ( 𝜑 → Ⅎ 𝑥 𝜓 )
4 dfrex2 ( ∃ 𝑦𝐴 𝜓 ↔ ¬ ∀ 𝑦𝐴 ¬ 𝜓 )
5 3 nfnd ( 𝜑 → Ⅎ 𝑥 ¬ 𝜓 )
6 1 2 5 nfrald ( 𝜑 → Ⅎ 𝑥𝑦𝐴 ¬ 𝜓 )
7 6 nfnd ( 𝜑 → Ⅎ 𝑥 ¬ ∀ 𝑦𝐴 ¬ 𝜓 )
8 4 7 nfxfrd ( 𝜑 → Ⅎ 𝑥𝑦𝐴 𝜓 )