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Theorem bnj1307 30852
Description: Technical lemma for bnj60 30891. 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
bnj1307.1 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1307.2 (𝑤𝐵 → ∀𝑥 𝑤𝐵)
Assertion
Ref Expression
bnj1307 (𝑤𝐶 → ∀𝑥 𝑤𝐶)
Distinct variable groups:   𝑤,𝐵   𝑤,𝑑,𝑥   𝑥,𝑓
Allowed substitution hints:   𝐵(𝑥,𝑓,𝑑)   𝐶(𝑥,𝑤,𝑓,𝑑)   𝐺(𝑥,𝑤,𝑓,𝑑)   𝑌(𝑥,𝑤,𝑓,𝑑)

Proof of Theorem bnj1307
StepHypRef Expression
1 bnj1307.1 . . 3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
2 bnj1307.2 . . . . . 6 (𝑤𝐵 → ∀𝑥 𝑤𝐵)
32nfcii 2752 . . . . 5 𝑥𝐵
4 nfv 1840 . . . . . 6 𝑥 𝑓 Fn 𝑑
5 nfra1 2937 . . . . . 6 𝑥𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)
64, 5nfan 1825 . . . . 5 𝑥(𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))
73, 6nfrex 3003 . . . 4 𝑥𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))
87nfab 2765 . . 3 𝑥{𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
91, 8nfcxfr 2759 . 2 𝑥𝐶
109nfcrii 2754 1 (𝑤𝐶 → ∀𝑥 𝑤𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1478   = wceq 1480  wcel 1987  {cab 2607  wral 2908  wrex 2909   Fn wfn 5852  cfv 5857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914
This theorem is referenced by:  bnj1311  30853  bnj1373  30859  bnj1498  30890  bnj1525  30898  bnj1523  30900
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