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Theorem eleqtrd 2429
Description: Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
Hypotheses
Ref Expression
eleqtrd.1 (φA B)
eleqtrd.2 (φB = C)
Assertion
Ref Expression
eleqtrd (φA C)

Proof of Theorem eleqtrd
StepHypRef Expression
1 eleqtrd.1 . 2 (φA B)
2 eleqtrd.2 . . 3 (φB = C)
32eleq2d 2420 . 2 (φ → (A BA C))
41, 3mpbid 201 1 (φA C)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1642   wcel 1710
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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  4451  tfinpw1  4494  sfintfin  4532  sfinltfin  4535  fnbr  5185  ecelqsdm  5994  enadjlem1  6059  nenpw1pwlem2  6085  spaccl  6286  fnfreclem3  6319  fnfrec  6320
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