Theorem nfsab 2185

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 1530	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3	2	hbab 2184	. 2 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} → ∀𝑥 𝑧 ∈ {𝑦 ∣ 𝜑})
4	3	nfi 1473	1 ⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜑}

Syntax hints:  Ⅎwnf 1471   ∈ wcel 2164  {cab 2179

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180

This theorem is referenced by:  nfab  2341  peano2  4627  lss1d  13879