Theorem lgricngricex

Description: There are two different locally isomorphic graphs which are not isomorphic. (Contributed by AV, 23-Nov-2025)

	Ref	Expression
Assertion	lgricngricex	$$⊢\exists g\exists h\left(g{\simeq }_{𝑙𝑔𝑟}h\wedge \neg g{\simeq }_{𝑔𝑟}h\right)$$

Proof

Step	Hyp	Ref	Expression
1		gpg5grlic	Could not format ( 5 gPetersenGr 1 ) ~=lgr ( 5 gPetersenGr 2 ) : No typesetting found for |- ( 5 gPetersenGr 1 ) ~=lgr ( 5 gPetersenGr 2 ) with typecode |-
2		gpg5ngric	Could not format -. ( 5 gPetersenGr 1 ) ~=gr ( 5 gPetersenGr 2 ) : No typesetting found for |- -. ( 5 gPetersenGr 1 ) ~=gr ( 5 gPetersenGr 2 ) with typecode |-
3		ovex	Could not format ( 5 gPetersenGr 1 ) e. _V : No typesetting found for |- ( 5 gPetersenGr 1 ) e. _V with typecode |-
4		ovex	Could not format ( 5 gPetersenGr 2 ) e. _V : No typesetting found for |- ( 5 gPetersenGr 2 ) e. _V with typecode |-
5		breq12	Could not format ( ( g = ( 5 gPetersenGr 1 ) /\ h = ( 5 gPetersenGr 2 ) ) -> ( g ~=lgr h <-> ( 5 gPetersenGr 1 ) ~=lgr ( 5 gPetersenGr 2 ) ) ) : No typesetting found for |- ( ( g = ( 5 gPetersenGr 1 ) /\ h = ( 5 gPetersenGr 2 ) ) -> ( g ~=lgr h <-> ( 5 gPetersenGr 1 ) ~=lgr ( 5 gPetersenGr 2 ) ) ) with typecode |-
6		breq12	Could not format ( ( g = ( 5 gPetersenGr 1 ) /\ h = ( 5 gPetersenGr 2 ) ) -> ( g ~=gr h <-> ( 5 gPetersenGr 1 ) ~=gr ( 5 gPetersenGr 2 ) ) ) : No typesetting found for |- ( ( g = ( 5 gPetersenGr 1 ) /\ h = ( 5 gPetersenGr 2 ) ) -> ( g ~=gr h <-> ( 5 gPetersenGr 1 ) ~=gr ( 5 gPetersenGr 2 ) ) ) with typecode |-
7	6	notbid	Could not format ( ( g = ( 5 gPetersenGr 1 ) /\ h = ( 5 gPetersenGr 2 ) ) -> ( -. g ~=gr h <-> -. ( 5 gPetersenGr 1 ) ~=gr ( 5 gPetersenGr 2 ) ) ) : No typesetting found for |- ( ( g = ( 5 gPetersenGr 1 ) /\ h = ( 5 gPetersenGr 2 ) ) -> ( -. g ~=gr h <-> -. ( 5 gPetersenGr 1 ) ~=gr ( 5 gPetersenGr 2 ) ) ) with typecode |-
8	5 7	anbi12d	Could not format ( ( g = ( 5 gPetersenGr 1 ) /\ h = ( 5 gPetersenGr 2 ) ) -> ( ( g ~=lgr h /\ -. g ~=gr h ) <-> ( ( 5 gPetersenGr 1 ) ~=lgr ( 5 gPetersenGr 2 ) /\ -. ( 5 gPetersenGr 1 ) ~=gr ( 5 gPetersenGr 2 ) ) ) ) : No typesetting found for |- ( ( g = ( 5 gPetersenGr 1 ) /\ h = ( 5 gPetersenGr 2 ) ) -> ( ( g ~=lgr h /\ -. g ~=gr h ) <-> ( ( 5 gPetersenGr 1 ) ~=lgr ( 5 gPetersenGr 2 ) /\ -. ( 5 gPetersenGr 1 ) ~=gr ( 5 gPetersenGr 2 ) ) ) ) with typecode |-
9	3 4 8	spc2ev	Could not format ( ( ( 5 gPetersenGr 1 ) ~=lgr ( 5 gPetersenGr 2 ) /\ -. ( 5 gPetersenGr 1 ) ~=gr ( 5 gPetersenGr 2 ) ) -> E. g E. h ( g ~=lgr h /\ -. g ~=gr h ) ) : No typesetting found for |- ( ( ( 5 gPetersenGr 1 ) ~=lgr ( 5 gPetersenGr 2 ) /\ -. ( 5 gPetersenGr 1 ) ~=gr ( 5 gPetersenGr 2 ) ) -> E. g E. h ( g ~=lgr h /\ -. g ~=gr h ) ) with typecode |-
10	1 2 9	mp2an	$$⊢\exists g\exists h\left(g{\simeq }_{𝑙𝑔𝑟}h\wedge \neg g{\simeq }_{𝑔𝑟}h\right)$$