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Metamath Proof Explorer

Theorem nfne

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

Ref Expression
Hypotheses nfne.1 ⊢ Ⅎ _ x A
nfne.2 ⊢ Ⅎ _ x B
Assertion nfne ⊢ Ⅎ x A ≠ B

Proof

Step Hyp Ref Expression
1 nfne.1 ⊢ Ⅎ _ x A
2 nfne.2 ⊢ Ⅎ _ x B
3 df-ne ⊢ A ≠ B ↔ ¬ A = B
4 1 2 nfeq ⊢ Ⅎ x A = B
5 4 nfn ⊢ Ⅎ x ¬ A = B
6 3 5 nfxfr ⊢ Ⅎ x A ≠ B