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Theorem vtocl2gaf 2614
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 ((x 𝐶 y 𝐷) → φ)
Assertion
Ref Expression
vtocl2gaf ((A 𝐶 B 𝐷) → 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 2185 . . . . 5 x A 𝐶
5 nfv 1418 . . . . 5 x y 𝐷
64, 5nfan 1454 . . . 4 x(A 𝐶 y 𝐷)
7 vtocl2gaf.1 . . . 4 xψ
86, 7nfim 1461 . . 3 x((A 𝐶 y 𝐷) → ψ)
92nfel1 2185 . . . . 5 y A 𝐶
103nfel1 2185 . . . . 5 y B 𝐷
119, 10nfan 1454 . . . 4 y(A 𝐶 B 𝐷)
12 vtocl2gaf.2 . . . 4 yχ
1311, 12nfim 1461 . . 3 y((A 𝐶 B 𝐷) → χ)
14 eleq1 2097 . . . . 5 (x = A → (x 𝐶A 𝐶))
1514anbi1d 438 . . . 4 (x = A → ((x 𝐶 y 𝐷) ↔ (A 𝐶 y 𝐷)))
16 vtocl2gaf.3 . . . 4 (x = A → (φψ))
1715, 16imbi12d 223 . . 3 (x = A → (((x 𝐶 y 𝐷) → φ) ↔ ((A 𝐶 y 𝐷) → ψ)))
18 eleq1 2097 . . . . 5 (y = B → (y 𝐷B 𝐷))
1918anbi2d 437 . . . 4 (y = B → ((A 𝐶 y 𝐷) ↔ (A 𝐶 B 𝐷)))
20 vtocl2gaf.4 . . . 4 (y = B → (ψχ))
2119, 20imbi12d 223 . . 3 (y = B → (((A 𝐶 y 𝐷) → ψ) ↔ ((A 𝐶 B 𝐷) → χ)))
22 vtocl2gaf.5 . . 3 ((x 𝐶 y 𝐷) → φ)
231, 2, 3, 8, 13, 17, 21, 22vtocl2gf 2609 . 2 ((A 𝐶 B 𝐷) → ((A 𝐶 B 𝐷) → χ))
2423pm2.43i 43 1 ((A 𝐶 B 𝐷) → χ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wnf 1346   wcel 1390  wnfc 2162
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553
This theorem is referenced by:  vtocl2ga  2615  ovmpt2s  5566  ov2gf  5567  ovi3  5579
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