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Theorem ceqsal 2884
 Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)
Hypotheses
Ref Expression
ceqsal.1 xψ
ceqsal.2 A V
ceqsal.3 (x = A → (φψ))
Assertion
Ref Expression
ceqsal (x(x = Aφ) ↔ ψ)
Distinct variable group:   x,A
Allowed substitution hints:   φ(x)   ψ(x)

Proof of Theorem ceqsal
StepHypRef Expression
1 ceqsal.2 . 2 A V
2 ceqsal.1 . . 3 xψ
3 ceqsal.3 . . 3 (x = A → (φψ))
42, 3ceqsalg 2883 . 2 (A V → (x(x = Aφ) ↔ ψ))
51, 4ax-mp 8 1 (x(x = Aφ) ↔ ψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Vcvv 2859 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-6 1729  ax-11 1746  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861 This theorem is referenced by:  ceqsalv  2885
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