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Theorem rabxfrd 4228
 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 𝑦𝐵
rabxfrd.2 𝑦𝐶
rabxfrd.3 ((𝜑𝑦𝐷) → 𝐴𝐷)
rabxfrd.4 (𝑥 = 𝐴 → (𝜓𝜒))
rabxfrd.5 (𝑦 = 𝐵𝐴 = 𝐶)
Assertion
Ref Expression
rabxfrd ((𝜑𝐵𝐷) → (𝐶 ∈ {𝑥𝐷𝜓} ↔ 𝐵 ∈ {𝑦𝐷𝜒}))
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
StepHypRef Expression
1 rabxfrd.3 . . . . . . . . . . 11 ((𝜑𝑦𝐷) → 𝐴𝐷)
21ex 112 . . . . . . . . . 10 (𝜑 → (𝑦𝐷𝐴𝐷))
3 ibibr 239 . . . . . . . . . 10 ((𝑦𝐷𝐴𝐷) ↔ (𝑦𝐷 → (𝐴𝐷𝑦𝐷)))
42, 3sylib 131 . . . . . . . . 9 (𝜑 → (𝑦𝐷 → (𝐴𝐷𝑦𝐷)))
54imp 119 . . . . . . . 8 ((𝜑𝑦𝐷) → (𝐴𝐷𝑦𝐷))
65anbi1d 446 . . . . . . 7 ((𝜑𝑦𝐷) → ((𝐴𝐷𝜒) ↔ (𝑦𝐷𝜒)))
7 rabxfrd.4 . . . . . . . 8 (𝑥 = 𝐴 → (𝜓𝜒))
87elrab 2720 . . . . . . 7 (𝐴 ∈ {𝑥𝐷𝜓} ↔ (𝐴𝐷𝜒))
9 rabid 2502 . . . . . . 7 (𝑦 ∈ {𝑦𝐷𝜒} ↔ (𝑦𝐷𝜒))
106, 8, 93bitr4g 216 . . . . . 6 ((𝜑𝑦𝐷) → (𝐴 ∈ {𝑥𝐷𝜓} ↔ 𝑦 ∈ {𝑦𝐷𝜒}))
1110rabbidva 2565 . . . . 5 (𝜑 → {𝑦𝐷𝐴 ∈ {𝑥𝐷𝜓}} = {𝑦𝐷𝑦 ∈ {𝑦𝐷𝜒}})
1211eleq2d 2123 . . . 4 (𝜑 → (𝐵 ∈ {𝑦𝐷𝐴 ∈ {𝑥𝐷𝜓}} ↔ 𝐵 ∈ {𝑦𝐷𝑦 ∈ {𝑦𝐷𝜒}}))
13 rabxfrd.1 . . . . 5 𝑦𝐵
14 nfcv 2194 . . . . 5 𝑦𝐷
15 rabxfrd.2 . . . . . 6 𝑦𝐶
1615nfel1 2204 . . . . 5 𝑦 𝐶 ∈ {𝑥𝐷𝜓}
17 rabxfrd.5 . . . . . 6 (𝑦 = 𝐵𝐴 = 𝐶)
1817eleq1d 2122 . . . . 5 (𝑦 = 𝐵 → (𝐴 ∈ {𝑥𝐷𝜓} ↔ 𝐶 ∈ {𝑥𝐷𝜓}))
1913, 14, 16, 18elrabf 2718 . . . 4 (𝐵 ∈ {𝑦𝐷𝐴 ∈ {𝑥𝐷𝜓}} ↔ (𝐵𝐷𝐶 ∈ {𝑥𝐷𝜓}))
20 nfrab1 2506 . . . . . 6 𝑦{𝑦𝐷𝜒}
2113, 20nfel 2202 . . . . 5 𝑦 𝐵 ∈ {𝑦𝐷𝜒}
22 eleq1 2116 . . . . 5 (𝑦 = 𝐵 → (𝑦 ∈ {𝑦𝐷𝜒} ↔ 𝐵 ∈ {𝑦𝐷𝜒}))
2313, 14, 21, 22elrabf 2718 . . . 4 (𝐵 ∈ {𝑦𝐷𝑦 ∈ {𝑦𝐷𝜒}} ↔ (𝐵𝐷𝐵 ∈ {𝑦𝐷𝜒}))
2412, 19, 233bitr3g 215 . . 3 (𝜑 → ((𝐵𝐷𝐶 ∈ {𝑥𝐷𝜓}) ↔ (𝐵𝐷𝐵 ∈ {𝑦𝐷𝜒})))
25 pm5.32 434 . . 3 ((𝐵𝐷 → (𝐶 ∈ {𝑥𝐷𝜓} ↔ 𝐵 ∈ {𝑦𝐷𝜒})) ↔ ((𝐵𝐷𝐶 ∈ {𝑥𝐷𝜓}) ↔ (𝐵𝐷𝐵 ∈ {𝑦𝐷𝜒})))
2624, 25sylibr 141 . 2 (𝜑 → (𝐵𝐷 → (𝐶 ∈ {𝑥𝐷𝜓} ↔ 𝐵 ∈ {𝑦𝐷𝜒})))
2726imp 119 1 ((𝜑𝐵𝐷) → (𝐶 ∈ {𝑥𝐷𝜓} ↔ 𝐵 ∈ {𝑦𝐷𝜒}))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ↔ wb 102   = wceq 1259   ∈ wcel 1409  Ⅎwnfc 2181  {crab 2327 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rab 2332  df-v 2576 This theorem is referenced by:  rabxfr  4229
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