Theorem sbcel12 4379

Description: Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Revised by NM, 18-Aug-2018.)

Assertion

Ref	Expression
sbcel12	⊢ ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶)

Proof of Theorem sbcel12

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 3750	. . . 4 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝐵 ∈ 𝐶 ↔ [𝐴 / 𝑥]𝐵 ∈ 𝐶))
2		dfsbcq2 3750	. . . . . 6 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝑦 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐵))
3	2	abbidv 2832	. . . . 5 ⊢ (𝑧 = 𝐴 → {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵})
4		dfsbcq2 3750	. . . . . 6 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝑦 ∈ 𝐶 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐶))
5	4	abbidv 2832	. . . . 5 ⊢ (𝑧 = 𝐴 → {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶})
6	3, 5	eleq12d 2860	. . . 4 ⊢ (𝑧 = 𝐴 → ({𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵} ∈ {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶} ↔ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ∈ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶}))
7		nfs1v 2194	. . . . . . 7 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝑦 ∈ 𝐵
8	7	nfab 2934	. . . . . 6 ⊢ Ⅎ𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵}
9		nfs1v 2194	. . . . . . 7 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝑦 ∈ 𝐶
10	9	nfab 2934	. . . . . 6 ⊢ Ⅎ𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶}
11	8, 10	nfel 2942	. . . . 5 ⊢ Ⅎ𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵} ∈ {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶}
12		sbab 2912	. . . . . 6 ⊢ (𝑥 = 𝑧 → 𝐵 = {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵})
13		sbab 2912	. . . . . 6 ⊢ (𝑥 = 𝑧 → 𝐶 = {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶})
14	12, 13	eleq12d 2860	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝐵 ∈ 𝐶 ↔ {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵} ∈ {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶}))
15	11, 14	sbiev 2350	. . . 4 ⊢ ([𝑧 / 𝑥]𝐵 ∈ 𝐶 ↔ {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵} ∈ {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶})
16	1, 6, 15	vtoclbg 3527	. . 3 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ∈ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶}))
17		df-csb 3857	. . . 4 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
18		df-csb 3857	. . . 4 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶}
19	17, 18	eleq12i 2859	. . 3 ⊢ (⦋𝐴 / 𝑥⦌𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶 ↔ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ∈ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶})
20	16, 19	bitr4di 292	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶))
21		sbcex 3757	. . . 4 ⊢ ([𝐴 / 𝑥]𝐵 ∈ 𝐶 → 𝐴 ∈ V)
22	21	con3i 155	. . 3 ⊢ (¬ 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝐵 ∈ 𝐶)
23		noel 4294	. . . 4 ⊢ ¬ ⦋𝐴 / 𝑥⦌𝐵 ∈ ∅
24		csbprc 4377	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐶 = ∅)
25	24	eleq2d 2852	. . . 4 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ∈ ∅))
26	23, 25	mtbiri 330	. . 3 ⊢ (¬ 𝐴 ∈ V → ¬ ⦋𝐴 / 𝑥⦌𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶)
27	22, 26	2falsed 379	. 2 ⊢ (¬ 𝐴 ∈ V → ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶))
28	20, 27	pm2.61i 184	1 ⊢ ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   = wceq 1570  [wsb 2099   ∈ wcel 2146  {cab 2744  Vcvv 3458  [wsbc 3747  ⦋csb 3856  ∅c0 4289

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738

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 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-nul 4290

This theorem is used by:  sbcnel12g  4382  sbcel1g  4384  sbcel2  4386  sbccsb2  4405  csbmpt12  5547  ixpsnval  8907  fmptdf2  33038  csbmpo123  38018  csbfinxpg  38075  finixpnum  38297  csbxpgVD  45643  csbrngVD  45645