Theorem eqelb

Description: Substitution of equal classes into elementhood relation. (Contributed by Peter Mazsa, 17-Jul-2019)

	Ref	Expression
Assertion	eqelb	|- ( ( A = B /\ A e. C ) <-> ( A = B /\ B e. C ) )

Proof

Step	Hyp	Ref	Expression
1		simpl	|- ( ( B = A /\ A e. C ) -> B = A )
2		eqeltr	|- ( ( B = A /\ A e. C ) -> B e. C )
3	1 2	jca	|- ( ( B = A /\ A e. C ) -> ( B = A /\ B e. C ) )
4		eqcom	|- ( B = A <-> A = B )
5	4	anbi1i	|- ( ( B = A /\ A e. C ) <-> ( A = B /\ A e. C ) )
6	4	anbi1i	|- ( ( B = A /\ B e. C ) <-> ( A = B /\ B e. C ) )
7	3 5 6	3imtr3i	|- ( ( A = B /\ A e. C ) -> ( A = B /\ B e. C ) )
8		simpl	|- ( ( A = B /\ B e. C ) -> A = B )
9		eqeltr	|- ( ( A = B /\ B e. C ) -> A e. C )
10	8 9	jca	|- ( ( A = B /\ B e. C ) -> ( A = B /\ A e. C ) )
11	7 10	impbii	|- ( ( A = B /\ A e. C ) <-> ( A = B /\ B e. C ) )