Theorem csbexga 4191

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 3105	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
2		abid2 2330	. . . . . . 7 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐵} = 𝐵
3		elex 2791	. . . . . . 7 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
4	2, 3	eqeltrid 2296	. . . . . 6 ⊢ (𝐵 ∈ 𝑊 → {𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V)
5	4	alimi 1481	. . . . 5 ⊢ (∀𝑥 𝐵 ∈ 𝑊 → ∀𝑥{𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V)
6		spsbc 3020	. . . . 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 2352	. . . . 5 ⊢ Ⅎ𝑥V
10	9	sbcabel 3091	. . . 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 2296	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 𝐵 ∈ 𝑊) → ⦋𝐴 / 𝑥⦌𝐵 ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1373   ∈ wcel 2180  {cab 2195  Vcvv 2779  [wsbc 3008  ⦋csb 3104

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-v 2781  df-sbc 3009  df-csb 3105

This theorem is referenced by:  csbexa  4192  prdsex  13268  imasex  13304