Theorem rabxfrd 5417

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 412	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ 𝐷 → 𝐴 ∈ 𝐷))
3		ibibr 368	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐷 → 𝐴 ∈ 𝐷) ↔ (𝑦 ∈ 𝐷 → (𝐴 ∈ 𝐷 ↔ 𝑦 ∈ 𝐷)))
4	2, 3	sylib 218	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ 𝐷 → (𝐴 ∈ 𝐷 ↔ 𝑦 ∈ 𝐷)))
5	4	imp 406	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → (𝐴 ∈ 𝐷 ↔ 𝑦 ∈ 𝐷))
6	5	anbi1d 631	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → ((𝐴 ∈ 𝐷 ∧ 𝜒) ↔ (𝑦 ∈ 𝐷 ∧ 𝜒)))
7		rabxfrd.4	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))
8	7	elrab 3692	. . . . . . 7 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓} ↔ (𝐴 ∈ 𝐷 ∧ 𝜒))
9		rabid 3458	. . . . . . 7 ⊢ (𝑦 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒} ↔ (𝑦 ∈ 𝐷 ∧ 𝜒))
10	6, 8, 9	3bitr4g 314	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → (𝐴 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓} ↔ 𝑦 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒}))
11	10	rabbidva 3443	. . . . 5 ⊢ (𝜑 → {𝑦 ∈ 𝐷 ∣ 𝐴 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓}} = {𝑦 ∈ 𝐷 ∣ 𝑦 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒}})
12	11	eleq2d 2827	. . . 4 ⊢ (𝜑 → (𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝐴 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓}} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒}}))
13		rabxfrd.1	. . . . 5 ⊢ Ⅎ𝑦𝐵
14		nfcv 2905	. . . . 5 ⊢ Ⅎ𝑦𝐷
15		rabxfrd.2	. . . . . 6 ⊢ Ⅎ𝑦𝐶
16	15	nfel1 2922	. . . . 5 ⊢ Ⅎ𝑦 𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓}
17		rabxfrd.5	. . . . . 6 ⊢ (𝑦 = 𝐵 → 𝐴 = 𝐶)
18	17	eleq1d 2826	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓} ↔ 𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓}))
19	13, 14, 16, 18	elrabf 3688	. . . 4 ⊢ (𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝐴 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓}} ↔ (𝐵 ∈ 𝐷 ∧ 𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓}))
20		nfrab1 3457	. . . . . 6 ⊢ Ⅎ𝑦{𝑦 ∈ 𝐷 ∣ 𝜒}
21	13, 20	nfel 2920	. . . . 5 ⊢ Ⅎ𝑦 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒}
22		eleq1 2829	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒}))
23	13, 14, 21, 22	elrabf 3688	. . . 4 ⊢ (𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒}} ↔ (𝐵 ∈ 𝐷 ∧ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒}))
24	12, 19, 23	3bitr3g 313	. . 3 ⊢ (𝜑 → ((𝐵 ∈ 𝐷 ∧ 𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓}) ↔ (𝐵 ∈ 𝐷 ∧ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒})))
25		pm5.32 573	. . 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 234	. 2 ⊢ (𝜑 → (𝐵 ∈ 𝐷 → (𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒})))
27	26	imp 406	1 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐷) → (𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Ⅎwnfc 2890  {crab 3436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-rab 3437  df-v 3482

This theorem is referenced by:  rabxfr  5418  riotaxfrd  7422