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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