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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 )