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Theorem dedhb 3639
Description: A deduction theorem for converting the inference 𝑥𝐴 => 𝜑 into a closed theorem. Use nfa1 2149 and nfab 2911 to eliminate the hypothesis of the substitution instance 𝜓 of the inference. For converting the inference form into a deduction form, abidnf 3638 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 3638 . . . 4 (𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧𝐴} = 𝐴)
32eqcomd 2743 . . 3 (𝑥𝐴𝐴 = {𝑧 ∣ ∀𝑥 𝑧𝐴})
4 dedhb.1 . . 3 (𝐴 = {𝑧 ∣ ∀𝑥 𝑧𝐴} → (𝜑𝜓))
53, 4syl 17 . 2 (𝑥𝐴 → (𝜑𝜓))
61, 5mpbiri 257 1 (𝑥𝐴𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1537   = wceq 1539  wcel 2107  {cab 2714  wnfc 2885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-12 2172  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887
This theorem is referenced by: (None)
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