Theorem eqeltr 34001

Description: Substitution of equal classes into elementhood relation. (Contributed by Peter Mazsa, 22-Jul-2017.)

Assertion

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

Proof of Theorem eqeltr

Step	Hyp	Ref	Expression
1		eleq1 2689	. 2 |- ( A = B -> ( A e. C <-> B e. C ) )
2	1	biimpar 502	1 |- ( ( A = B /\ B e. C ) -> A e. C )

Syntax hints:   -> wi 4   /\ 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:  eqelb  34002