Theorem rabxfrd 5310

Description: Membership in a restricted class abstraction after substituting an expression 𝐴 (containing 𝑦) for 𝑥 in the formula defining the class abstraction. (Contributed by NM, 16-Jan-2012.)

Hypotheses

Ref	Expression
rabxfrd.1	⊢ Ⅎ𝑦𝐵
rabxfrd.2	⊢ Ⅎ𝑦𝐶
rabxfrd.3	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → 𝐴 ∈ 𝐷)
rabxfrd.4	⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))
rabxfrd.5	⊢ (𝑦 = 𝐵 → 𝐴 = 𝐶)

Assertion

Ref	Expression
rabxfrd	⊢ ((𝜑 ∧ 𝐵 ∈ 𝐷) → (𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒}))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐷   𝜑,𝑦   𝜓,𝑦   𝜒,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝜒(𝑦)   𝐴(𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem rabxfrd

Step	Hyp	Ref	Expression
1		rabxfrd.3	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → 𝐴 ∈ 𝐷)
2	1	ex 416	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ 𝐷 → 𝐴 ∈ 𝐷))
3		ibibr 372	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐷 → 𝐴 ∈ 𝐷) ↔ (𝑦 ∈ 𝐷 → (𝐴 ∈ 𝐷 ↔ 𝑦 ∈ 𝐷)))
4	2, 3	sylib 221	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ 𝐷 → (𝐴 ∈ 𝐷 ↔ 𝑦 ∈ 𝐷)))
5	4	imp 410	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → (𝐴 ∈ 𝐷 ↔ 𝑦 ∈ 𝐷))
6	5	anbi1d 633	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → ((𝐴 ∈ 𝐷 ∧ 𝜒) ↔ (𝑦 ∈ 𝐷 ∧ 𝜒)))
7		rabxfrd.4	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))
8	7	elrab 3602	. . . . . . 7 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓} ↔ (𝐴 ∈ 𝐷 ∧ 𝜒))
9		rabid 3290	. . . . . . 7 ⊢ (𝑦 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒} ↔ (𝑦 ∈ 𝐷 ∧ 𝜒))
10	6, 8, 9	3bitr4g 317	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → (𝐴 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓} ↔ 𝑦 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒}))
11	10	rabbidva 3388	. . . . 5 ⊢ (𝜑 → {𝑦 ∈ 𝐷 ∣ 𝐴 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓}} = {𝑦 ∈ 𝐷 ∣ 𝑦 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒}})
12	11	eleq2d 2823	. . . 4 ⊢ (𝜑 → (𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝐴 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓}} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒}}))
13		rabxfrd.1	. . . . 5 ⊢ Ⅎ𝑦𝐵
14		nfcv 2904	. . . . 5 ⊢ Ⅎ𝑦𝐷
15		rabxfrd.2	. . . . . 6 ⊢ Ⅎ𝑦𝐶
16	15	nfel1 2920	. . . . 5 ⊢ Ⅎ𝑦 𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓}
17		rabxfrd.5	. . . . . 6 ⊢ (𝑦 = 𝐵 → 𝐴 = 𝐶)
18	17	eleq1d 2822	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓} ↔ 𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓}))
19	13, 14, 16, 18	elrabf 3598	. . . 4 ⊢ (𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝐴 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓}} ↔ (𝐵 ∈ 𝐷 ∧ 𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓}))
20		nfrab1 3296	. . . . . 6 ⊢ Ⅎ𝑦{𝑦 ∈ 𝐷 ∣ 𝜒}
21	13, 20	nfel 2918	. . . . 5 ⊢ Ⅎ𝑦 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒}
22		eleq1 2825	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒}))
23	13, 14, 21, 22	elrabf 3598	. . . 4 ⊢ (𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒}} ↔ (𝐵 ∈ 𝐷 ∧ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒}))
24	12, 19, 23	3bitr3g 316	. . 3 ⊢ (𝜑 → ((𝐵 ∈ 𝐷 ∧ 𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓}) ↔ (𝐵 ∈ 𝐷 ∧ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒})))
25		pm5.32 577	. . 3 ⊢ ((𝐵 ∈ 𝐷 → (𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒})) ↔ ((𝐵 ∈ 𝐷 ∧ 𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓}) ↔ (𝐵 ∈ 𝐷 ∧ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒})))
26	24, 25	sylibr 237	. 2 ⊢ (𝜑 → (𝐵 ∈ 𝐷 → (𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒})))
27	26	imp 410	1 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐷) → (𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒}))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2110  Ⅎwnfc 2884  {crab 3065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-rab 3070  df-v 3410

This theorem is referenced by:  rabxfr  5311  riotaxfrd  7205