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Theorem ceqsralv 2886
 Description: Restricted quantifier version of ceqsalv 2885. (Contributed by NM, 21-Jun-2013.)
Hypothesis
Ref Expression
ceqsralv.2 (x = A → (φψ))
Assertion
Ref Expression
ceqsralv (A B → (x B (x = Aφ) ↔ ψ))
Distinct variable groups:   x,A   x,B   ψ,x
Allowed substitution hint:   φ(x)

Proof of Theorem ceqsralv
StepHypRef Expression
1 nfv 1619 . 2 xψ
2 ceqsralv.2 . . 3 (x = A → (φψ))
32ax-gen 1546 . 2 x(x = A → (φψ))
4 ceqsralt 2882 . 2 ((Ⅎxψ x(x = A → (φψ)) A B) → (x B (x = Aφ) ↔ ψ))
51, 3, 4mp3an12 1267 1 (A B → (x B (x = Aφ) ↔ ψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  ∀wral 2614 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-6 1729  ax-11 1746  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-ral 2619  df-v 2861 This theorem is referenced by:  eqreu  3028
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