Metamath Proof Explorer


Theorem nfan1c

Description: Variant of nfan and commuted form of nfan1 . (Contributed by BTernaryTau, 31-Jul-2025)

Ref Expression
Hypotheses nfan1c.1 𝑥 𝜑
nfan1c.2 ( 𝜑 → Ⅎ 𝑥 𝜓 )
Assertion nfan1c 𝑥 ( 𝜓𝜑 )

Proof

Step Hyp Ref Expression
1 nfan1c.1 𝑥 𝜑
2 nfan1c.2 ( 𝜑 → Ⅎ 𝑥 𝜓 )
3 1 2 nfan1 𝑥 ( 𝜑𝜓 )
4 ancom ( ( 𝜑𝜓 ) ↔ ( 𝜓𝜑 ) )
5 4 nfbii ( Ⅎ 𝑥 ( 𝜑𝜓 ) ↔ Ⅎ 𝑥 ( 𝜓𝜑 ) )
6 3 5 mpbi 𝑥 ( 𝜓𝜑 )