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Theorem bnj1133 33658
Description: Technical lemma for bnj69 33679. 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 33646 . . 3 (𝑛𝐷 → E Fr 𝑛)
41, 3bnj769 33431 . 2 (𝜒 → E Fr 𝑛)
5 impexp 452 . . . . . 6 (((𝑖𝑛𝜏) → 𝜃) ↔ (𝑖𝑛 → (𝜏𝜃)))
65bicomi 223 . . . . 5 ((𝑖𝑛 → (𝜏𝜃)) ↔ ((𝑖𝑛𝜏) → 𝜃))
76albii 1822 . . . 4 (∀𝑖(𝑖𝑛 → (𝜏𝜃)) ↔ ∀𝑖((𝑖𝑛𝜏) → 𝜃))
8 bnj1133.8 . . . 4 ((𝑖𝑛𝜏) → 𝜃)
97, 8mpgbir 1802 . . 3 𝑖(𝑖𝑛 → (𝜏𝜃))
10 df-ral 3062 . . 3 (∀𝑖𝑛 (𝜏𝜃) ↔ ∀𝑖(𝑖𝑛 → (𝜏𝜃)))
119, 10mpbir 230 . 2 𝑖𝑛 (𝜏𝜃)
12 vex 3448 . . 3 𝑛 ∈ V
13 bnj1133.7 . . 3 (𝜏 ↔ ∀𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜃))
1412, 13bnj110 33527 . 2 (( E Fr 𝑛 ∧ ∀𝑖𝑛 (𝜏𝜃)) → ∀𝑖𝑛 𝜃)
154, 11, 14sylancl 587 1 (𝜒 → ∀𝑖𝑛 𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540   = wceq 1542  wcel 2107  wral 3061  [wsbc 3740  cdif 3908  c0 4283  {csn 4587   class class class wbr 5106   E cep 5537   Fr wfr 5586   Fn wfn 6492  ωcom 7803  w-bnj17 33355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-tr 5224  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-ord 6321  df-on 6322  df-om 7804  df-bnj17 33356
This theorem is referenced by:  bnj1128  33659
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