Theorem eqeltrrd 2428

Description: Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)

Hypotheses

Ref	Expression
eqeltrrd.1	⊢ (φ → A = B)
eqeltrrd.2	⊢ (φ → A ∈ C)

Assertion

Ref	Expression
eqeltrrd	⊢ (φ → B ∈ C)

Proof of Theorem eqeltrrd

Step	Hyp	Ref	Expression
1		eqeltrrd.1	. . 3 ⊢ (φ → A = B)
2	1	eqcomd 2358	. 2 ⊢ (φ → B = A)
3		eqeltrrd.2	. 2 ⊢ (φ → A ∈ C)
4	2, 3	eqeltrd 2427	1 ⊢ (φ → B ∈ C)

Syntax hints:   → wi 4   = 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:  3eltr3d  2433  nnsucelr  4429  sfinltfin  4536  vfin1cltv  4548  vfinspsslem1  4551  phi11lem1  4596  xpexr2  5111  ffvresb  5432  ncdisjun  6137  eqtc  6162