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Theorem vtocl2gaf 2637
Description: Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 10-Aug-2013.)
Hypotheses
Ref Expression
vtocl2gaf.a 𝑥𝐴
vtocl2gaf.b 𝑦𝐴
vtocl2gaf.c 𝑦𝐵
vtocl2gaf.1 𝑥𝜓
vtocl2gaf.2 𝑦𝜒
vtocl2gaf.3 (𝑥 = 𝐴 → (𝜑𝜓))
vtocl2gaf.4 (𝑦 = 𝐵 → (𝜓𝜒))
vtocl2gaf.5 ((𝑥𝐶𝑦𝐷) → 𝜑)
Assertion
Ref Expression
vtocl2gaf ((𝐴𝐶𝐵𝐷) → 𝜒)
Distinct variable groups:   𝑥,𝑦,𝐶   𝑥,𝐷,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem vtocl2gaf
StepHypRef Expression
1 vtocl2gaf.a . . 3 𝑥𝐴
2 vtocl2gaf.b . . 3 𝑦𝐴
3 vtocl2gaf.c . . 3 𝑦𝐵
41nfel1 2204 . . . . 5 𝑥 𝐴𝐶
5 nfv 1437 . . . . 5 𝑥 𝑦𝐷
64, 5nfan 1473 . . . 4 𝑥(𝐴𝐶𝑦𝐷)
7 vtocl2gaf.1 . . . 4 𝑥𝜓
86, 7nfim 1480 . . 3 𝑥((𝐴𝐶𝑦𝐷) → 𝜓)
92nfel1 2204 . . . . 5 𝑦 𝐴𝐶
103nfel1 2204 . . . . 5 𝑦 𝐵𝐷
119, 10nfan 1473 . . . 4 𝑦(𝐴𝐶𝐵𝐷)
12 vtocl2gaf.2 . . . 4 𝑦𝜒
1311, 12nfim 1480 . . 3 𝑦((𝐴𝐶𝐵𝐷) → 𝜒)
14 eleq1 2116 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐶𝐴𝐶))
1514anbi1d 446 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐶𝑦𝐷) ↔ (𝐴𝐶𝑦𝐷)))
16 vtocl2gaf.3 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
1715, 16imbi12d 227 . . 3 (𝑥 = 𝐴 → (((𝑥𝐶𝑦𝐷) → 𝜑) ↔ ((𝐴𝐶𝑦𝐷) → 𝜓)))
18 eleq1 2116 . . . . 5 (𝑦 = 𝐵 → (𝑦𝐷𝐵𝐷))
1918anbi2d 445 . . . 4 (𝑦 = 𝐵 → ((𝐴𝐶𝑦𝐷) ↔ (𝐴𝐶𝐵𝐷)))
20 vtocl2gaf.4 . . . 4 (𝑦 = 𝐵 → (𝜓𝜒))
2119, 20imbi12d 227 . . 3 (𝑦 = 𝐵 → (((𝐴𝐶𝑦𝐷) → 𝜓) ↔ ((𝐴𝐶𝐵𝐷) → 𝜒)))
22 vtocl2gaf.5 . . 3 ((𝑥𝐶𝑦𝐷) → 𝜑)
231, 2, 3, 8, 13, 17, 21, 22vtocl2gf 2632 . 2 ((𝐴𝐶𝐵𝐷) → ((𝐴𝐶𝐵𝐷) → 𝜒))
2423pm2.43i 47 1 ((𝐴𝐶𝐵𝐷) → 𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wnf 1365  wcel 1409  wnfc 2181
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-v 2576
This theorem is referenced by:  vtocl2ga  2638  ovmpt2s  5651  ov2gf  5652  ovi3  5664
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