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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