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Metamath Proof Explorer

Theorem nfcd

Description: Deduce that a class A does not have x free in it. (Contributed by Mario Carneiro, 11-Aug-2016)

Ref Expression
Hypotheses nfcd.1 ⊢ Ⅎ 𝑦 𝜑
nfcd.2 ⊢ ( 𝜑 → Ⅎ 𝑥 𝑦 ∈ 𝐴 )
Assertion nfcd ( 𝜑 → Ⅎ 𝑥 𝐴 )

Proof

Step Hyp Ref Expression
1 nfcd.1 ⊢ Ⅎ 𝑦 𝜑
2 nfcd.2 ⊢ ( 𝜑 → Ⅎ 𝑥 𝑦 ∈ 𝐴 )
3 1 2 alrimi ⊢ ( 𝜑 → ∀ 𝑦 Ⅎ 𝑥 𝑦 ∈ 𝐴 )
4 df-nfc ⊢ ( Ⅎ 𝑥 𝐴 ↔ ∀ 𝑦 Ⅎ 𝑥 𝑦 ∈ 𝐴 )
5 3 4 sylibr ⊢ ( 𝜑 → Ⅎ 𝑥 𝐴 )