Metamath Proof Explorer


Theorem nfabd

Description: Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker nfabdw when possible. (Contributed by Mario Carneiro, 8-Oct-2016) Avoid ax-9 and ax-ext . (Revised by Wolf Lammen, 23-May-2023) (New usage is discouraged.)

Ref Expression
Hypotheses nfabd.1 𝑦 𝜑
nfabd.2 ( 𝜑 → Ⅎ 𝑥 𝜓 )
Assertion nfabd ( 𝜑 𝑥 { 𝑦𝜓 } )

Proof

Step Hyp Ref Expression
1 nfabd.1 𝑦 𝜑
2 nfabd.2 ( 𝜑 → Ⅎ 𝑥 𝜓 )
3 nfv 𝑧 𝜑
4 df-clab ( 𝑧 ∈ { 𝑦𝜓 } ↔ [ 𝑧 / 𝑦 ] 𝜓 )
5 1 2 nfsbd ( 𝜑 → Ⅎ 𝑥 [ 𝑧 / 𝑦 ] 𝜓 )
6 4 5 nfxfrd ( 𝜑 → Ⅎ 𝑥 𝑧 ∈ { 𝑦𝜓 } )
7 3 6 nfcd ( 𝜑 𝑥 { 𝑦𝜓 } )