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Metamath Proof Explorer

Theorem nfcsbd

Description: Deduction version of nfcsb . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 21-Nov-2005) (Revised by Mario Carneiro, 12-Oct-2016) (New usage is discouraged.)

Ref Expression
Hypotheses nfcsbd.1 ⊢ Ⅎ y φ
nfcsbd.2 ⊢ φ → Ⅎ _ x A
nfcsbd.3 ⊢ φ → Ⅎ _ x B
Assertion nfcsbd ⊢ φ → Ⅎ _ x ⦋ A / y⦌ B

Proof

Step Hyp Ref Expression
1 nfcsbd.1 ⊢ Ⅎ y φ
2 nfcsbd.2 ⊢ φ → Ⅎ _ x A
3 nfcsbd.3 ⊢ φ → Ⅎ _ x B
4 df-csb ⊢ ⦋ A / y⦌ B = z | [˙A / y]˙ z ∈ B
5 nfv ⊢ Ⅎ z φ
6 3 nfcrd ⊢ φ → Ⅎ x z ∈ B
7 1 2 6 nfsbcd ⊢ φ → Ⅎ x [˙A / y]˙ z ∈ B
8 5 7 nfabd ⊢ φ → Ⅎ _ x z | [˙A / y]˙ z ∈ B
9 4 8 nfcxfrd ⊢ φ → Ⅎ _ x ⦋ A / y⦌ B