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Theorem bnj1083 31374
Description: Technical lemma for bnj69 31406. 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 3056 . 2 (∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ↔ ∃𝑛(𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)))
2 bnj1083.8 . . 3 𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
32abeq2i 2873 . 2 (𝑓𝐾 ↔ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓))
4 bnj1083.3 . . . 4 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
5 bnj252 31099 . . . 4 ((𝑛𝐷𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)))
64, 5bitri 264 . . 3 (𝜒 ↔ (𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)))
76exbii 1923 . 2 (∃𝑛𝜒 ↔ ∃𝑛(𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)))
81, 3, 73bitr4i 292 1 (𝑓𝐾 ↔ ∃𝑛𝜒)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  w3a 1072   = wceq 1632  wex 1853  wcel 2139  {cab 2746  wrex 3051   Fn wfn 6044  w-bnj17 31082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-12 2196  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-rex 3056  df-bnj17 31083
This theorem is referenced by:  bnj1121  31381  bnj1145  31389
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