Theorem nfcsb1d 3167

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 3138	. 2 ⊢ [A / x]B = {y ∣ [̣A / x]̣y ∈ B}
2		nfv 1619	. . 3 ⊢ Ⅎyφ
3		nfcsb1d.1	. . . 4 ⊢ (φ → ℲxA)
4	3	nfsbc1d 3064	. . 3 ⊢ (φ → Ⅎx[̣A / x]̣y ∈ B)
5	2, 4	nfabd 2509	. 2 ⊢ (φ → Ⅎx{y ∣ [̣A / x]̣y ∈ B})
6	1, 5	nfcxfrd 2488	1 ⊢ (φ → Ⅎx[A / x]B)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2477  [̣wsbc 3047  [csb 3137

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-sbc 3048  df-csb 3138

This theorem is used by:  nfcsb1  3168