Theorem nfcsb1d 3158

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 3128	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
2		nfv 1576	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfcsb1d.1	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
4	3	nfsbc1d 3048	. . 3 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑥]𝑦 ∈ 𝐵)
5	2, 4	nfabd 2394	. 2 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵})
6	1, 5	nfcxfrd 2372	1 ⊢ (𝜑 → Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵)

Syntax hints:   → wi 4   ∈ wcel 2202  {cab 2217  Ⅎwnfc 2361  [wsbc 3031  ⦋csb 3127

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-sbc 3032  df-csb 3128

This theorem is referenced by:  nfcsb1  3159  riotaeqimp  5995