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Theorem dedhb 2704
Description: A deduction theorem for converting the inference xA => φ into a closed theorem. Use nfa1 1431 and nfab 2179 to eliminate the hypothesis of the substitution instance ψ of the inference. For converting the inference form into a deduction form, abidnf 2703 is useful. (Contributed by NM, 8-Dec-2006.)
Hypotheses
Ref Expression
dedhb.1 (A = {zx 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 2703 . . . 4 (xA → {zx z A} = A)
32eqcomd 2042 . . 3 (xAA = {zx z A})
4 dedhb.1 . . 3 (A = {zx z A} → (φψ))
53, 4syl 14 . 2 (xA → (φψ))
61, 5mpbiri 157 1 (xAφ)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1240   = wceq 1242   wcel 1390  {cab 2023  wnfc 2162
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164
This theorem is referenced by: (None)
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