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Theorem dedhb 3006
 Description: A deduction theorem for converting the inference ⊢ ℲxA => ⊢ φ into a closed theorem. Use nfa1 1788 and nfab 2493 to eliminate the hypothesis of the substitution instance ψ of the inference. For converting the inference form into a deduction form, abidnf 3005 is useful. (Contributed by NM, 8-Dec-2006.)
Hypotheses
Ref Expression
dedhb.1 (A = {z x z A} → (φψ))
dedhb.2 ψ
Assertion
Ref Expression
dedhb (xAφ)
Distinct variable groups:   x,z   z,A
Allowed substitution hints:   φ(x,z)   ψ(x,z)   A(x)

Proof of Theorem dedhb
StepHypRef Expression
1 dedhb.2 . 2 ψ
2 abidnf 3005 . . . 4 (xA → {z x z A} = A)
32eqcomd 2358 . . 3 (xAA = {z x z A})
4 dedhb.1 . . 3 (A = {z x z A} → (φψ))
53, 4syl 15 . 2 (xA → (φψ))
61, 5mpbiri 224 1 (xAφ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2476 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-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 This theorem is referenced by: (None)
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