Theorem nfabd 2274

Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.)

Hypotheses

Ref	Expression
nfabd.1	⊢ Ⅎ𝑦𝜑
nfabd.2	⊢ (𝜑 → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfabd	⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})

Proof of Theorem nfabd

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1491	. 2 ⊢ Ⅎ𝑧𝜑
2		df-clab 2102	. . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓)
3		nfabd.1	. . . 4 ⊢ Ⅎ𝑦𝜑
4		nfabd.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
5	3, 4	nfsbd 1926	. . 3 ⊢ (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓)
6	2, 5	nfxfrd 1434	. 2 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜓})
7	1, 6	nfcd 2250	1 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})

Syntax hints:   → wi 4  Ⅎwnf 1419   ∈ wcel 1463  [wsb 1718  {cab 2101  Ⅎwnfc 2242

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498

This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-nfc 2244

This theorem is referenced by:  nfsbcd  2897  nfcsb1d  2999  nfcsbd  3002  nfifd  3465  nfunid  3709  nfiotadxy  5049  nfixpxy  6565