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Theorem rabxfrd 5017
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 397 . . . . . . . . . 10 (𝜑 → (𝑦𝐷𝐴𝐷))
3 ibibr 357 . . . . . . . . . 10 ((𝑦𝐷𝐴𝐷) ↔ (𝑦𝐷 → (𝐴𝐷𝑦𝐷)))
42, 3sylib 208 . . . . . . . . 9 (𝜑 → (𝑦𝐷 → (𝐴𝐷𝑦𝐷)))
54imp 393 . . . . . . . 8 ((𝜑𝑦𝐷) → (𝐴𝐷𝑦𝐷))
65anbi1d 607 . . . . . . 7 ((𝜑𝑦𝐷) → ((𝐴𝐷𝜒) ↔ (𝑦𝐷𝜒)))
7 rabxfrd.4 . . . . . . . 8 (𝑥 = 𝐴 → (𝜓𝜒))
87elrab 3513 . . . . . . 7 (𝐴 ∈ {𝑥𝐷𝜓} ↔ (𝐴𝐷𝜒))
9 rabid 3263 . . . . . . 7 (𝑦 ∈ {𝑦𝐷𝜒} ↔ (𝑦𝐷𝜒))
106, 8, 93bitr4g 303 . . . . . 6 ((𝜑𝑦𝐷) → (𝐴 ∈ {𝑥𝐷𝜓} ↔ 𝑦 ∈ {𝑦𝐷𝜒}))
1110rabbidva 3337 . . . . 5 (𝜑 → {𝑦𝐷𝐴 ∈ {𝑥𝐷𝜓}} = {𝑦𝐷𝑦 ∈ {𝑦𝐷𝜒}})
1211eleq2d 2835 . . . 4 (𝜑 → (𝐵 ∈ {𝑦𝐷𝐴 ∈ {𝑥𝐷𝜓}} ↔ 𝐵 ∈ {𝑦𝐷𝑦 ∈ {𝑦𝐷𝜒}}))
13 rabxfrd.1 . . . . 5 𝑦𝐵
14 nfcv 2912 . . . . 5 𝑦𝐷
15 rabxfrd.2 . . . . . 6 𝑦𝐶
1615nfel1 2927 . . . . 5 𝑦 𝐶 ∈ {𝑥𝐷𝜓}
17 rabxfrd.5 . . . . . 6 (𝑦 = 𝐵𝐴 = 𝐶)
1817eleq1d 2834 . . . . 5 (𝑦 = 𝐵 → (𝐴 ∈ {𝑥𝐷𝜓} ↔ 𝐶 ∈ {𝑥𝐷𝜓}))
1913, 14, 16, 18elrabf 3509 . . . 4 (𝐵 ∈ {𝑦𝐷𝐴 ∈ {𝑥𝐷𝜓}} ↔ (𝐵𝐷𝐶 ∈ {𝑥𝐷𝜓}))
20 nfrab1 3270 . . . . . 6 𝑦{𝑦𝐷𝜒}
2113, 20nfel 2925 . . . . 5 𝑦 𝐵 ∈ {𝑦𝐷𝜒}
22 eleq1 2837 . . . . 5 (𝑦 = 𝐵 → (𝑦 ∈ {𝑦𝐷𝜒} ↔ 𝐵 ∈ {𝑦𝐷𝜒}))
2313, 14, 21, 22elrabf 3509 . . . 4 (𝐵 ∈ {𝑦𝐷𝑦 ∈ {𝑦𝐷𝜒}} ↔ (𝐵𝐷𝐵 ∈ {𝑦𝐷𝜒}))
2412, 19, 233bitr3g 302 . . 3 (𝜑 → ((𝐵𝐷𝐶 ∈ {𝑥𝐷𝜓}) ↔ (𝐵𝐷𝐵 ∈ {𝑦𝐷𝜒})))
25 pm5.32 555 . . 3 ((𝐵𝐷 → (𝐶 ∈ {𝑥𝐷𝜓} ↔ 𝐵 ∈ {𝑦𝐷𝜒})) ↔ ((𝐵𝐷𝐶 ∈ {𝑥𝐷𝜓}) ↔ (𝐵𝐷𝐵 ∈ {𝑦𝐷𝜒})))
2624, 25sylibr 224 . 2 (𝜑 → (𝐵𝐷 → (𝐶 ∈ {𝑥𝐷𝜓} ↔ 𝐵 ∈ {𝑦𝐷𝜒})))
2726imp 393 1 ((𝜑𝐵𝐷) → (𝐶 ∈ {𝑥𝐷𝜓} ↔ 𝐵 ∈ {𝑦𝐷𝜒}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wcel 2144  wnfc 2899  {crab 3064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rab 3069  df-v 3351
This theorem is referenced by:  rabxfr  5018  riotaxfrd  6784
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