Theorem rabxfrd 4447

Description: Class builder membership after substituting an expression 𝐴 (containing 𝑦) for 𝑥 in the class expression 𝜒. (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 114	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ 𝐷 → 𝐴 ∈ 𝐷))
3		ibibr 245	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐷 → 𝐴 ∈ 𝐷) ↔ (𝑦 ∈ 𝐷 → (𝐴 ∈ 𝐷 ↔ 𝑦 ∈ 𝐷)))
4	2, 3	sylib 121	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ 𝐷 → (𝐴 ∈ 𝐷 ↔ 𝑦 ∈ 𝐷)))
5	4	imp 123	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → (𝐴 ∈ 𝐷 ↔ 𝑦 ∈ 𝐷))
6	5	anbi1d 461	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → ((𝐴 ∈ 𝐷 ∧ 𝜒) ↔ (𝑦 ∈ 𝐷 ∧ 𝜒)))
7		rabxfrd.4	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))
8	7	elrab 2882	. . . . . . 7 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓} ↔ (𝐴 ∈ 𝐷 ∧ 𝜒))
9		rabid 2641	. . . . . . 7 ⊢ (𝑦 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒} ↔ (𝑦 ∈ 𝐷 ∧ 𝜒))
10	6, 8, 9	3bitr4g 222	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → (𝐴 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓} ↔ 𝑦 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒}))
11	10	rabbidva 2714	. . . . 5 ⊢ (𝜑 → {𝑦 ∈ 𝐷 ∣ 𝐴 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓}} = {𝑦 ∈ 𝐷 ∣ 𝑦 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒}})
12	11	eleq2d 2236	. . . 4 ⊢ (𝜑 → (𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝐴 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓}} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒}}))
13		rabxfrd.1	. . . . 5 ⊢ Ⅎ𝑦𝐵
14		nfcv 2308	. . . . 5 ⊢ Ⅎ𝑦𝐷
15		rabxfrd.2	. . . . . 6 ⊢ Ⅎ𝑦𝐶
16	15	nfel1 2319	. . . . 5 ⊢ Ⅎ𝑦 𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓}
17		rabxfrd.5	. . . . . 6 ⊢ (𝑦 = 𝐵 → 𝐴 = 𝐶)
18	17	eleq1d 2235	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓} ↔ 𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓}))
19	13, 14, 16, 18	elrabf 2880	. . . 4 ⊢ (𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝐴 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓}} ↔ (𝐵 ∈ 𝐷 ∧ 𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓}))
20		nfrab1 2645	. . . . . 6 ⊢ Ⅎ𝑦{𝑦 ∈ 𝐷 ∣ 𝜒}
21	13, 20	nfel 2317	. . . . 5 ⊢ Ⅎ𝑦 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒}
22		eleq1 2229	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒}))
23	13, 14, 21, 22	elrabf 2880	. . . 4 ⊢ (𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒}} ↔ (𝐵 ∈ 𝐷 ∧ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒}))
24	12, 19, 23	3bitr3g 221	. . 3 ⊢ (𝜑 → ((𝐵 ∈ 𝐷 ∧ 𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓}) ↔ (𝐵 ∈ 𝐷 ∧ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒})))
25		pm5.32 449	. . 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 133	. 2 ⊢ (𝜑 → (𝐵 ∈ 𝐷 → (𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒})))
27	26	imp 123	1 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐷) → (𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒}))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1343   ∈ wcel 2136  Ⅎwnfc 2295  {crab 2448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rab 2453  df-v 2728

This theorem is referenced by:  rabxfr  4448