Theorem eleq1i 2416

Description: Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
eleq1i.1	⊢ A = B

Assertion

Ref	Expression
eleq1i	⊢ (A ∈ C ↔ B ∈ C)

Proof of Theorem eleq1i

Step	Hyp	Ref	Expression
1		eleq1i.1	. 2 ⊢ A = B
2		eleq1 2413	. 2 ⊢ (A = B → (A ∈ C ↔ B ∈ C))
3	1, 2	ax-mp 5	1 ⊢ (A ∈ C ↔ B ∈ C)

Syntax hints:   ↔ wb 176   = wceq 1642   ∈ wcel 1710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-clel 2349

This theorem is referenced by:  eleq12i  2418  eqeltri  2423  opeqexb  4621  ssrel  4845  proj1eldm  4928  co01  5094  fressnfv  5440