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Metamath Proof Explorer

Theorem nfcsbw

Description: Bound-variable hypothesis builder for substitution into a class. Version of nfcsb with a disjoint variable condition, which does not require ax-13 . (Contributed by Mario Carneiro, 12-Oct-2016) Avoid ax-13 . (Revised by GG, 10-Jan-2024)

Ref Expression
Hypotheses nfcsbw.1 ⊢ Ⅎ _ x A
nfcsbw.2 ⊢ Ⅎ _ x B
Assertion nfcsbw ⊢ Ⅎ _ x ⦋ A / y⦌ B

Proof

Step Hyp Ref Expression
1 nfcsbw.1 ⊢ Ⅎ _ x A
2 nfcsbw.2 ⊢ Ⅎ _ x B
3 df-csb ⊢ ⦋ A / y⦌ B = z | [˙A / y]˙ z ∈ B
4 nftru ⊢ Ⅎ z ⊤
5 nftru ⊢ Ⅎ y ⊤
6 1 a1i ⊢ ⊤ → Ⅎ _ x A
7 2 a1i ⊢ ⊤ → Ⅎ _ x B
8 7 nfcrd ⊢ ⊤ → Ⅎ x z ∈ B
9 5 6 8 nfsbcdw ⊢ ⊤ → Ⅎ x [˙A / y]˙ z ∈ B
10 4 9 nfabdw ⊢ ⊤ → Ⅎ _ x z | [˙A / y]˙ z ∈ B
11 3 10 nfcxfrd ⊢ ⊤ → Ⅎ _ x ⦋ A / y⦌ B
12 11 mptru ⊢ Ⅎ _ x ⦋ A / y⦌ B