Theorem nfsab 2188

Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)

Hypothesis

Ref	Expression
nfsab.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
nfsab	⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜑}

Distinct variable group:   𝑥,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem nfsab

Step	Hyp	Ref	Expression
1		nfsab.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2	1	nfri 1533	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3	2	hbab 2187	. 2 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} → ∀𝑥 𝑧 ∈ {𝑦 ∣ 𝜑})
4	3	nfi 1476	1 ⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜑}

Syntax hints:  Ⅎwnf 1474   ∈ wcel 2167  {cab 2182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183

This theorem is referenced by:  nfab  2344  peano2  4631  lss1d  13939