Theorem csbexg 5263

Description: The existence of proper substitution into a class. (Contributed by NM, 10-Nov-2005.) (Revised by NM, 17-Aug-2018.)

Assertion

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

Proof of Theorem csbexg

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-csb 3847	. . 3 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
2		abid2 2897	. . . . . . . 8 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐵} = 𝐵
3		elex 3471	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
4	2, 3	eqeltrid 2864	. . . . . . 7 ⊢ (𝐵 ∈ 𝑊 → {𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V)
5	4	alimi 1844	. . . . . 6 ⊢ (∀𝑥 𝐵 ∈ 𝑊 → ∀𝑥{𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V)
6		spsbc 3751	. . . . . 6 ⊢ (𝐴 ∈ V → (∀𝑥{𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V → [𝐴 / 𝑥]{𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V))
7	5, 6	syl5 35	. . . . 5 ⊢ (𝐴 ∈ V → (∀𝑥 𝐵 ∈ 𝑊 → [𝐴 / 𝑥]{𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V))
8		nfcv 2922	. . . . . 6 ⊢ Ⅎ𝑥V
9	8	sbcabel 3824	. . . . 5 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]{𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V ↔ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ∈ V))
10	7, 9	sylibd 242	. . . 4 ⊢ (𝐴 ∈ V → (∀𝑥 𝐵 ∈ 𝑊 → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ∈ V))
11	10	imp 412	. . 3 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 𝐵 ∈ 𝑊) → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ∈ V)
12	1, 11	eqeltrid 2864	. 2 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 𝐵 ∈ 𝑊) → ⦋𝐴 / 𝑥⦌𝐵 ∈ V)
13		csbprc 4366	. . . 4 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)
14		0ex 5260	. . . 4 ⊢ ∅ ∈ V
15	13, 14	eqeltrdi 2868	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 ∈ V)
16	15	adantr 486	. 2 ⊢ ((¬ 𝐴 ∈ V ∧ ∀𝑥 𝐵 ∈ 𝑊) → ⦋𝐴 / 𝑥⦌𝐵 ∈ V)
17	12, 16	pm2.61ian 824	1 ⊢ (∀𝑥 𝐵 ∈ 𝑊 → ⦋𝐴 / 𝑥⦌𝐵 ∈ V)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401  ∀wal 1568   ∈ wcel 2145  {cab 2738  Vcvv 3450  [wsbc 3738  ⦋csb 3846  ∅c0 4278

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-nul 4279

This theorem is used by:  csbex  5264  abfmpeld  33181