Theorem nfcsb1d 3155

Description: Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)

Hypothesis

Ref	Expression
nfcsb1d.1	⊢ (𝜑 → Ⅎ𝑥𝐴)

Assertion

Ref	Expression
nfcsb1d	⊢ (𝜑 → Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵)

Proof of Theorem nfcsb1d

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-csb 3125	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
2		nfv 1574	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfcsb1d.1	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
4	3	nfsbc1d 3045	. . 3 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑥]𝑦 ∈ 𝐵)
5	2, 4	nfabd 2392	. 2 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵})
6	1, 5	nfcxfrd 2370	1 ⊢ (𝜑 → Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵)

Syntax hints:   → wi 4   ∈ wcel 2200  {cab 2215  Ⅎwnfc 2359  [wsbc 3028  ⦋csb 3124

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-sbc 3029  df-csb 3125

This theorem is referenced by:  nfcsb1  3156  riotaeqimp  5979