Theorem eleqtrrd 2430

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

Hypotheses

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

Assertion

Ref	Expression
eleqtrrd	⊢ (φ → A ∈ C)

Proof of Theorem eleqtrrd

Step	Hyp	Ref	Expression
1		eleqtrrd.1	. 2 ⊢ (φ → A ∈ B)
2		eleqtrrd.2	. . 3 ⊢ (φ → C = B)
3	2	eqcomd 2358	. 2 ⊢ (φ → B = C)
4	1, 3	eleqtrd 2429	1 ⊢ (φ → A ∈ 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:  3eltr4d  2434  tfin11  4494  tfinnn  4535  fvopab4t  5386  elimdelov  5574  enadjlem1  6060  enprmaplem3  6079  pw1eltc  6163  spacid  6286  spaccl  6287  frecsuc  6323