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Metamath Proof Explorer

Theorem nfeq

Description: Hypothesis builder for equality. (Contributed by NM, 21-Jun-1993) (Revised by Mario Carneiro, 11-Aug-2016) (Proof shortened by Wolf Lammen, 16-Nov-2019)

Ref Expression
Hypotheses nfnfc.1 ⊢ Ⅎ _ x A
nfeq.2 ⊢ Ⅎ _ x B
Assertion nfeq ⊢ Ⅎ x A = B

Proof

Step Hyp Ref Expression
1 nfnfc.1 ⊢ Ⅎ _ x A
2 nfeq.2 ⊢ Ⅎ _ x B
3 1 a1i ⊢ ⊤ → Ⅎ _ x A
4 2 a1i ⊢ ⊤ → Ⅎ _ x B
5 3 4 nfeqd ⊢ ⊤ → Ⅎ x A = B
6 5 mptru ⊢ Ⅎ x A = B