Theorem eleqtrd 2429

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

Hypotheses

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

Assertion

Ref	Expression
eleqtrd	⊢ (φ → A ∈ C)

Proof of Theorem eleqtrd

Step	Hyp	Ref	Expression
1		eleqtrd.1	. 2 ⊢ (φ → A ∈ B)
2		eleqtrd.2	. . 3 ⊢ (φ → B = C)
3	2	eleq2d 2420	. 2 ⊢ (φ → (A ∈ B ↔ A ∈ C))
4	1, 3	mpbid 201	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:  eleqtrrd  2430  3eltr3d  2433  syl5eleq  2439  syl6eleq  2443  prepeano4  4452  tfinpw1  4495  sfintfin  4533  sfinltfin  4536  fnbr  5186  ecelqsdm  5995  enadjlem1  6060  nenpw1pwlem2  6086  spaccl  6287  fnfreclem3  6320  fnfrec  6321