Theorem eqelb 34002

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

Assertion

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

Proof of Theorem eqelb

Step	Hyp	Ref	Expression
1		simpl 473	. . . 4 |- ( ( B = A /\ A e. C ) -> B = A )
2		eqeltr 34001	. . . 4 |- ( ( B = A /\ A e. C ) -> B e. C )
3	1, 2	jca 554	. . 3 |- ( ( B = A /\ A e. C ) -> ( B = A /\ B e. C ) )
4		eqcom 2629	. . . 4 |- ( B = A <-> A = B )
5	4	anbi1i 731	. . 3 |- ( ( B = A /\ A e. C ) <-> ( A = B /\ A e. C ) )
6	4	anbi1i 731	. . 3 |- ( ( B = A /\ B e. C ) <-> ( A = B /\ B e. C ) )
7	3, 5, 6	3imtr3i 280	. 2 |- ( ( A = B /\ A e. C ) -> ( A = B /\ B e. C ) )
8		simpl 473	. . 3 |- ( ( A = B /\ B e. C ) -> A = B )
9		eqeltr 34001	. . 3 |- ( ( A = B /\ B e. C ) -> A e. C )
10	8, 9	jca 554	. 2 |- ( ( A = B /\ B e. C ) -> ( A = B /\ A e. C ) )
11	7, 10	impbii 199	1 |- ( ( A = B /\ A e. C ) <-> ( A = B /\ B e. C ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615  df-clel 2618

This theorem is referenced by: (None)