Theorem csbexg 5316

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 3909	. . 3 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
2		abid2 2877	. . . . . . . 8 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐵} = 𝐵
3		elex 3499	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
4	2, 3	eqeltrid 2843	. . . . . . 7 ⊢ (𝐵 ∈ 𝑊 → {𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V)
5	4	alimi 1808	. . . . . 6 ⊢ (∀𝑥 𝐵 ∈ 𝑊 → ∀𝑥{𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V)
6		spsbc 3804	. . . . . 6 ⊢ (𝐴 ∈ V → (∀𝑥{𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V → [𝐴 / 𝑥]{𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V))
7	5, 6	syl5 34	. . . . 5 ⊢ (𝐴 ∈ V → (∀𝑥 𝐵 ∈ 𝑊 → [𝐴 / 𝑥]{𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V))
8		nfcv 2903	. . . . . 6 ⊢ Ⅎ𝑥V
9	8	sbcabel 3887	. . . . 5 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]{𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V ↔ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ∈ V))
10	7, 9	sylibd 239	. . . 4 ⊢ (𝐴 ∈ V → (∀𝑥 𝐵 ∈ 𝑊 → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ∈ V))
11	10	imp 406	. . 3 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 𝐵 ∈ 𝑊) → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ∈ V)
12	1, 11	eqeltrid 2843	. 2 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 𝐵 ∈ 𝑊) → ⦋𝐴 / 𝑥⦌𝐵 ∈ V)
13		csbprc 4415	. . . 4 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)
14		0ex 5313	. . . 4 ⊢ ∅ ∈ V
15	13, 14	eqeltrdi 2847	. . 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 812	1 ⊢ (∀𝑥 𝐵 ∈ 𝑊 → ⦋𝐴 / 𝑥⦌𝐵 ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1535   ∈ wcel 2106  {cab 2712  Vcvv 3478  [wsbc 3791  ⦋csb 3908  ∅c0 4339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-nul 5312

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-nul 4340

This theorem is referenced by:  csbex  5317  abfmpeld  32671