Theorem nfcsb1d 3062

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 3032	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
2		nfv 1508	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfcsb1d.1	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
4	3	nfsbc1d 2953	. . 3 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑥]𝑦 ∈ 𝐵)
5	2, 4	nfabd 2319	. 2 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵})
6	1, 5	nfcxfrd 2297	1 ⊢ (𝜑 → Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵)

Syntax hints:   → wi 4   ∈ wcel 2128  {cab 2143  Ⅎwnfc 2286  [wsbc 2937  ⦋csb 3031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-sbc 2938  df-csb 3032

This theorem is referenced by:  nfcsb1  3063