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Theorem bnj1083 32364
Description: Technical lemma for bnj69 32396. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1083.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj1083.8 𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
Assertion
Ref Expression
bnj1083 (𝑓𝐾 ↔ ∃𝑛𝜒)

Proof of Theorem bnj1083
StepHypRef Expression
1 df-rex 3115 . 2 (∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ↔ ∃𝑛(𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)))
2 bnj1083.8 . . 3 𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
32abeq2i 2928 . 2 (𝑓𝐾 ↔ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓))
4 bnj1083.3 . . . 4 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
5 bnj252 32087 . . . 4 ((𝑛𝐷𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)))
64, 5bitri 278 . . 3 (𝜒 ↔ (𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)))
76exbii 1849 . 2 (∃𝑛𝜒 ↔ ∃𝑛(𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)))
81, 3, 73bitr4i 306 1 (𝑓𝐾 ↔ ∃𝑛𝜒)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  w3a 1084   = wceq 1538  wex 1781  wcel 2112  {cab 2779  wrex 3110   Fn wfn 6323  w-bnj17 32070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-rex 3115  df-bnj17 32071
This theorem is referenced by:  bnj1121  32371  bnj1145  32379
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