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Theorem ceqsexgv 2971
 Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 29-Dec-1996.)
Hypothesis
Ref Expression
ceqsexgv.1 (x = A → (φψ))
Assertion
Ref Expression
ceqsexgv (A V → (x(x = A φ) ↔ ψ))
Distinct variable groups:   x,A   ψ,x
Allowed substitution hints:   φ(x)   V(x)

Proof of Theorem ceqsexgv
StepHypRef Expression
1 nfv 1619 . 2 xψ
2 ceqsexgv.1 . 2 (x = A → (φψ))
31, 2ceqsexg 2970 1 (A V → (x(x = A φ) ↔ ψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = 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-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861 This theorem is referenced by:  ceqsrexv  2972  clel3g  2976  eluni1g  4172  opkelopkabg  4245  otkelins2kg  4253  otkelins3kg  4254  opkelcokg  4261
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