Theorem csbexga 4218

Description: The existence of proper substitution into a class. (Contributed by NM, 10-Nov-2005.)

Assertion

Ref	Expression
csbexga	⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 𝐵 ∈ 𝑊) → ⦋𝐴 / 𝑥⦌𝐵 ∈ V)

Proof of Theorem csbexga

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-csb 3127	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
2		abid2 2351	. . . . . . 7 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐵} = 𝐵
3		elex 2813	. . . . . . 7 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
4	2, 3	eqeltrid 2317	. . . . . 6 ⊢ (𝐵 ∈ 𝑊 → {𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V)
5	4	alimi 1503	. . . . 5 ⊢ (∀𝑥 𝐵 ∈ 𝑊 → ∀𝑥{𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V)
6		spsbc 3042	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥{𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V → [𝐴 / 𝑥]{𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V))
7	5, 6	syl5 32	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 𝐵 ∈ 𝑊 → [𝐴 / 𝑥]{𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V))
8	7	imp 124	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 𝐵 ∈ 𝑊) → [𝐴 / 𝑥]{𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V)
9		nfcv 2373	. . . . 5 ⊢ Ⅎ𝑥V
10	9	sbcabel 3113	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]{𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V ↔ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ∈ V))
11	10	adantr 276	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 𝐵 ∈ 𝑊) → ([𝐴 / 𝑥]{𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V ↔ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ∈ V))
12	8, 11	mpbid 147	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 𝐵 ∈ 𝑊) → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ∈ V)
13	1, 12	eqeltrid 2317	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 𝐵 ∈ 𝑊) → ⦋𝐴 / 𝑥⦌𝐵 ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1395   ∈ wcel 2201  {cab 2216  Vcvv 2801  [wsbc 3030  ⦋csb 3126

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 2212

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-v 2803  df-sbc 3031  df-csb 3127

This theorem is referenced by:  csbexa  4219  prdsex  13375  imasex  13411