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Metamath Proof Explorer

Theorem nfned

Description: Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007) (Revised by Mario Carneiro, 7-Oct-2016)

Ref Expression
Hypotheses nfned.1 ⊢ ( 𝜑 → Ⅎ 𝑥 𝐴 )
nfned.2 ⊢ ( 𝜑 → Ⅎ 𝑥 𝐵 )
Assertion nfned ( 𝜑 → Ⅎ 𝑥 𝐴 ≠ 𝐵 )

Proof

Step Hyp Ref Expression
1 nfned.1 ⊢ ( 𝜑 → Ⅎ 𝑥 𝐴 )
2 nfned.2 ⊢ ( 𝜑 → Ⅎ 𝑥 𝐵 )
3 df-ne ⊢ ( 𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵 )
4 1 2 nfeqd ⊢ ( 𝜑 → Ⅎ 𝑥 𝐴 = 𝐵 )
5 4 nfnd ⊢ ( 𝜑 → Ⅎ 𝑥 ¬ 𝐴 = 𝐵 )
6 3 5 nfxfrd ⊢ ( 𝜑 → Ⅎ 𝑥 𝐴 ≠ 𝐵 )