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Theorem bnj1133 32346
Description: Technical lemma for bnj69 32367. 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 32334 . . 3 (𝑛𝐷 → E Fr 𝑛)
41, 3bnj769 32118 . 2 (𝜒 → E Fr 𝑛)
5 impexp 454 . . . . . 6 (((𝑖𝑛𝜏) → 𝜃) ↔ (𝑖𝑛 → (𝜏𝜃)))
65bicomi 227 . . . . 5 ((𝑖𝑛 → (𝜏𝜃)) ↔ ((𝑖𝑛𝜏) → 𝜃))
76albii 1821 . . . 4 (∀𝑖(𝑖𝑛 → (𝜏𝜃)) ↔ ∀𝑖((𝑖𝑛𝜏) → 𝜃))
8 bnj1133.8 . . . 4 ((𝑖𝑛𝜏) → 𝜃)
97, 8mpgbir 1801 . . 3 𝑖(𝑖𝑛 → (𝜏𝜃))
10 df-ral 3138 . . 3 (∀𝑖𝑛 (𝜏𝜃) ↔ ∀𝑖(𝑖𝑛 → (𝜏𝜃)))
119, 10mpbir 234 . 2 𝑖𝑛 (𝜏𝜃)
12 vex 3483 . . 3 𝑛 ∈ V
13 bnj1133.7 . . 3 (𝜏 ↔ ∀𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜃))
1412, 13bnj110 32215 . 2 (( E Fr 𝑛 ∧ ∀𝑖𝑛 (𝜏𝜃)) → ∀𝑖𝑛 𝜃)
154, 11, 14sylancl 589 1 (𝜒 → ∀𝑖𝑛 𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536   = wceq 1538  wcel 2115  wral 3133  [wsbc 3758  cdif 3916  c0 4276  {csn 4550   class class class wbr 5053   E cep 5452   Fr wfr 5499   Fn wfn 6340  ωcom 7576  w-bnj17 32041
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318  ax-un 7457
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-br 5054  df-opab 5116  df-tr 5160  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-ord 6183  df-on 6184  df-lim 6185  df-suc 6186  df-om 7577  df-bnj17 32042
This theorem is referenced by:  bnj1128  32347
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