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Theorem eleq2s 2445
 Description: Substitution of equal classes into a membership antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
Hypotheses
Ref Expression
eleq2s.1 (A Bφ)
eleq2s.2 C = B
Assertion
Ref Expression
eleq2s (A Cφ)

Proof of Theorem eleq2s
StepHypRef Expression
1 eleq2s.2 . . 3 C = B
21eleq2i 2417 . 2 (A CA B)
3 eleq2s.1 . 2 (A Bφ)
42, 3sylbi 187 1 (A Cφ)
 Colors of variables: wff setvar class 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:  elxpi  4800  optocl  4838  ecexr  5950  ectocld  5991  ecoptocl  5996  nulnnc  6118  ncprc  6124  elnc  6125
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