Theorem nfcsb1d 3089

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 3059	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
2		nfv 1528	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfcsb1d.1	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
4	3	nfsbc1d 2980	. . 3 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑥]𝑦 ∈ 𝐵)
5	2, 4	nfabd 2339	. 2 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵})
6	1, 5	nfcxfrd 2317	1 ⊢ (𝜑 → Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵)

Syntax hints:   → wi 4   ∈ wcel 2148  {cab 2163  Ⅎwnfc 2306  [wsbc 2963  ⦋csb 3058

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-sbc 2964  df-csb 3059

This theorem is referenced by:  nfcsb1  3090