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Metamath Proof Explorer

Theorem vtocl2gafOLD

Description: Obsolete version of vtocl2gaf as of 31-May-2025. (Contributed by NM, 10-Aug-2013) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses vtocl2gaf.a 
|- F/_ x A
vtocl2gaf.b 
|- F/_ y A
vtocl2gaf.c 
|- F/_ y B
vtocl2gaf.1 
|- F/ x ps
vtocl2gaf.2 
|- F/ y ch
vtocl2gaf.3 
|- ( x = A -> ( ph <-> ps ) )
vtocl2gaf.4 
|- ( y = B -> ( ps <-> ch ) )
vtocl2gaf.5 
|- ( ( x e. C /\ y e. D ) -> ph )
Assertion vtocl2gafOLD 
|- ( ( A e. C /\ B e. D ) -> ch )

Proof

Step Hyp Ref Expression
1 vtocl2gaf.a 
 |-  F/_ x A
2 vtocl2gaf.b 
 |-  F/_ y A
3 vtocl2gaf.c 
 |-  F/_ y B
4 vtocl2gaf.1 
 |-  F/ x ps
5 vtocl2gaf.2 
 |-  F/ y ch
6 vtocl2gaf.3 
 |-  ( x = A -> ( ph <-> ps ) )
7 vtocl2gaf.4 
 |-  ( y = B -> ( ps <-> ch ) )
8 vtocl2gaf.5 
 |-  ( ( x e. C /\ y e. D ) -> ph )
9 1 nfel1 
 |-  F/ x A e. C
10 nfv 
 |-  F/ x y e. D
11 9 10 nfan 
 |-  F/ x ( A e. C /\ y e. D )
12 11 4 nfim 
 |-  F/ x ( ( A e. C /\ y e. D ) -> ps )
13 2 nfel1 
 |-  F/ y A e. C
14 3 nfel1 
 |-  F/ y B e. D
15 13 14 nfan 
 |-  F/ y ( A e. C /\ B e. D )
16 15 5 nfim 
 |-  F/ y ( ( A e. C /\ B e. D ) -> ch )
17 eleq1 
 |-  ( x = A -> ( x e. C <-> A e. C ) )
18 17 anbi1d 
 |-  ( x = A -> ( ( x e. C /\ y e. D ) <-> ( A e. C /\ y e. D ) ) )
19 18 6 imbi12d 
 |-  ( x = A -> ( ( ( x e. C /\ y e. D ) -> ph ) <-> ( ( A e. C /\ y e. D ) -> ps ) ) )
20 eleq1 
 |-  ( y = B -> ( y e. D <-> B e. D ) )
21 20 anbi2d 
 |-  ( y = B -> ( ( A e. C /\ y e. D ) <-> ( A e. C /\ B e. D ) ) )
22 21 7 imbi12d 
 |-  ( y = B -> ( ( ( A e. C /\ y e. D ) -> ps ) <-> ( ( A e. C /\ B e. D ) -> ch ) ) )
23 1 2 3 12 16 19 22 8 vtocl2gf 
 |-  ( ( A e. C /\ B e. D ) -> ( ( A e. C /\ B e. D ) -> ch ) )
24 23 pm2.43i 
 |-  ( ( A e. C /\ B e. D ) -> ch )