Theorem csbexg 4720

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 3500	. . 3 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
2		abid2 2732	. . . . . . . 8 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐵} = 𝐵
3		elex 3185	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
4	2, 3	syl5eqel 2692	. . . . . . 7 ⊢ (𝐵 ∈ 𝑊 → {𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V)
5	4	alimi 1730	. . . . . 6 ⊢ (∀𝑥 𝐵 ∈ 𝑊 → ∀𝑥{𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V)
6		spsbc 3415	. . . . . 6 ⊢ (𝐴 ∈ V → (∀𝑥{𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V → [𝐴 / 𝑥]{𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V))
7	5, 6	syl5 33	. . . . 5 ⊢ (𝐴 ∈ V → (∀𝑥 𝐵 ∈ 𝑊 → [𝐴 / 𝑥]{𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V))
8		nfcv 2751	. . . . . 6 ⊢ Ⅎ𝑥V
9	8	sbcabel 3483	. . . . 5 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]{𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V ↔ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ∈ V))
10	7, 9	sylibd 228	. . . 4 ⊢ (𝐴 ∈ V → (∀𝑥 𝐵 ∈ 𝑊 → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ∈ V))
11	10	imp 444	. . 3 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 𝐵 ∈ 𝑊) → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ∈ V)
12	1, 11	syl5eqel 2692	. 2 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 𝐵 ∈ 𝑊) → ⦋𝐴 / 𝑥⦌𝐵 ∈ V)
13		csbprc 3932	. . . 4 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)
14		0ex 4718	. . . 4 ⊢ ∅ ∈ V
15	13, 14	syl6eqel 2696	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 ∈ V)
16	15	adantr 480	. 2 ⊢ ((¬ 𝐴 ∈ V ∧ ∀𝑥 𝐵 ∈ 𝑊) → ⦋𝐴 / 𝑥⦌𝐵 ∈ V)
17	12, 16	pm2.61ian 827	1 ⊢ (∀𝑥 𝐵 ∈ 𝑊 → ⦋𝐴 / 𝑥⦌𝐵 ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383  ∀wal 1473   ∈ wcel 1977  {cab 2596  Vcvv 3173  [wsbc 3402  ⦋csb 3499  ∅c0 3874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-nul 3875

This theorem is referenced by:  csbex  4721  abfmpeld  28834