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Theorem dedhb 2733
 Description: A deduction theorem for converting the inference ⊢ Ⅎ𝑥𝐴 => ⊢ 𝜑 into a closed theorem. Use nfa1 1450 and nfab 2198 to eliminate the hypothesis of the substitution instance 𝜓 of the inference. For converting the inference form into a deduction form, abidnf 2732 is useful. (Contributed by NM, 8-Dec-2006.)
Hypotheses
Ref Expression
dedhb.1 (𝐴 = {𝑧 ∣ ∀𝑥 𝑧𝐴} → (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 2732 . . . 4 (𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧𝐴} = 𝐴)
32eqcomd 2061 . . 3 (𝑥𝐴𝐴 = {𝑧 ∣ ∀𝑥 𝑧𝐴})
4 dedhb.1 . . 3 (𝐴 = {𝑧 ∣ ∀𝑥 𝑧𝐴} → (𝜑𝜓))
53, 4syl 14 . 2 (𝑥𝐴 → (𝜑𝜓))
61, 5mpbiri 161 1 (𝑥𝐴𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 102  ∀wal 1257   = wceq 1259   ∈ wcel 1409  {cab 2042  Ⅎwnfc 2181 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183 This theorem is referenced by: (None)
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