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Theorem dedhb 2942
Description: A deduction theorem for converting the inference 𝑥𝐴 => 𝜑 into a closed theorem. Use nfa1 1564 and nfab 2353 to eliminate the hypothesis of the substitution instance 𝜓 of the inference. For converting the inference form into a deduction form, abidnf 2941 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 2941 . . . 4 (𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧𝐴} = 𝐴)
32eqcomd 2211 . . 3 (𝑥𝐴𝐴 = {𝑧 ∣ ∀𝑥 𝑧𝐴})
4 dedhb.1 . . 3 (𝐴 = {𝑧 ∣ ∀𝑥 𝑧𝐴} → (𝜑𝜓))
53, 4syl 14 . 2 (𝑥𝐴 → (𝜑𝜓))
61, 5mpbiri 168 1 (𝑥𝐴𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wal 1371   = wceq 1373  wcel 2176  {cab 2191  wnfc 2335
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187
This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337
This theorem is referenced by: (None)
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