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Metamath Proof Explorer

Theorem nfdm

Description: Bound-variable hypothesis builder for domain. (Contributed by NM, 30-Jan-2004) (Revised by Mario Carneiro, 15-Oct-2016)

Ref Expression
Hypothesis nfrn.1 ⊢ Ⅎ _ x A
Assertion nfdm ⊢ Ⅎ _ x dom ⁡ A

Proof

Step Hyp Ref Expression
1 nfrn.1 ⊢ Ⅎ _ x A
2 df-dm ⊢ dom ⁡ A = y | ∃ z y A z
3 nfcv ⊢ Ⅎ _ x y
4 nfcv ⊢ Ⅎ _ x z
5 3 1 4 nfbr ⊢ Ⅎ x y A z
6 5 nfex ⊢ Ⅎ x ∃ z y A z
7 6 nfab ⊢ Ⅎ _ x y | ∃ z y A z
8 2 7 nfcxfr ⊢ Ⅎ _ x dom ⁡ A