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Theorem bnj1133 34982
Description: Technical lemma for bnj69 35003. 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
bnj1133.3 𝐷 = (ω ∖ {∅})
bnj1133.5 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj1133.7 (𝜏 ↔ ∀𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜃))
bnj1133.8 ((𝑖𝑛𝜏) → 𝜃)
Assertion
Ref Expression
bnj1133 (𝜒 → ∀𝑖𝑛 𝜃)
Distinct variable groups:   𝑖,𝑗,𝑛   𝜃,𝑗
Allowed substitution hints:   𝜑(𝑓,𝑖,𝑗,𝑛)   𝜓(𝑓,𝑖,𝑗,𝑛)   𝜒(𝑓,𝑖,𝑗,𝑛)   𝜃(𝑓,𝑖,𝑛)   𝜏(𝑓,𝑖,𝑗,𝑛)   𝐷(𝑓,𝑖,𝑗,𝑛)

Proof of Theorem bnj1133
StepHypRef Expression
1 bnj1133.5 . . 3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
2 bnj1133.3 . . . 4 𝐷 = (ω ∖ {∅})
32bnj1071 34970 . . 3 (𝑛𝐷 → E Fr 𝑛)
41, 3bnj769 34755 . 2 (𝜒 → E Fr 𝑛)
5 impexp 450 . . . . . 6 (((𝑖𝑛𝜏) → 𝜃) ↔ (𝑖𝑛 → (𝜏𝜃)))
65bicomi 224 . . . . 5 ((𝑖𝑛 → (𝜏𝜃)) ↔ ((𝑖𝑛𝜏) → 𝜃))
76albii 1816 . . . 4 (∀𝑖(𝑖𝑛 → (𝜏𝜃)) ↔ ∀𝑖((𝑖𝑛𝜏) → 𝜃))
8 bnj1133.8 . . . 4 ((𝑖𝑛𝜏) → 𝜃)
97, 8mpgbir 1796 . . 3 𝑖(𝑖𝑛 → (𝜏𝜃))
10 df-ral 3060 . . 3 (∀𝑖𝑛 (𝜏𝜃) ↔ ∀𝑖(𝑖𝑛 → (𝜏𝜃)))
119, 10mpbir 231 . 2 𝑖𝑛 (𝜏𝜃)
12 vex 3482 . . 3 𝑛 ∈ V
13 bnj1133.7 . . 3 (𝜏 ↔ ∀𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜃))
1412, 13bnj110 34851 . 2 (( E Fr 𝑛 ∧ ∀𝑖𝑛 (𝜏𝜃)) → ∀𝑖𝑛 𝜃)
154, 11, 14sylancl 586 1 (𝜒 → ∀𝑖𝑛 𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  wcel 2106  wral 3059  [wsbc 3791  cdif 3960  c0 4339  {csn 4631   class class class wbr 5148   E cep 5588   Fr wfr 5638   Fn wfn 6558  ωcom 7887  w-bnj17 34679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-tr 5266  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390  df-om 7888  df-bnj17 34680
This theorem is referenced by:  bnj1128  34983
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