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Theorem ceqsalg 2883
 Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 29-Oct-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Hypotheses
Ref Expression
ceqsalg.1 xψ
ceqsalg.2 (x = A → (φψ))
Assertion
Ref Expression
ceqsalg (A V → (x(x = Aφ) ↔ ψ))
Distinct variable group:   x,A
Allowed substitution hints:   φ(x)   ψ(x)   V(x)

Proof of Theorem ceqsalg
StepHypRef Expression
1 elisset 2869 . . 3 (A Vx x = A)
2 nfa1 1788 . . . 4 xx(x = Aφ)
3 ceqsalg.1 . . . 4 xψ
4 ceqsalg.2 . . . . . . 7 (x = A → (φψ))
54biimpd 198 . . . . . 6 (x = A → (φψ))
65a2i 12 . . . . 5 ((x = Aφ) → (x = Aψ))
76sps 1754 . . . 4 (x(x = Aφ) → (x = Aψ))
82, 3, 7exlimd 1806 . . 3 (x(x = Aφ) → (x x = Aψ))
91, 8syl5com 26 . 2 (A V → (x(x = Aφ) → ψ))
104biimprcd 216 . . 3 (ψ → (x = Aφ))
113, 10alrimi 1765 . 2 (ψx(x = Aφ))
129, 11impbid1 194 1 (A V → (x(x = Aφ) ↔ ψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544   = 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-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:  ceqsal  2884  sbc6g  3071  uniiunlem  3353
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