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Theorem bnj1133 31362
Description: Technical lemma for bnj69 31383. 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 31350 . . 3 (𝑛𝐷 → E Fr 𝑛)
41, 3bnj769 31137 . 2 (𝜒 → E Fr 𝑛)
5 impexp 461 . . . . . 6 (((𝑖𝑛𝜏) → 𝜃) ↔ (𝑖𝑛 → (𝜏𝜃)))
65bicomi 214 . . . . 5 ((𝑖𝑛 → (𝜏𝜃)) ↔ ((𝑖𝑛𝜏) → 𝜃))
76albii 1894 . . . 4 (∀𝑖(𝑖𝑛 → (𝜏𝜃)) ↔ ∀𝑖((𝑖𝑛𝜏) → 𝜃))
8 bnj1133.8 . . . 4 ((𝑖𝑛𝜏) → 𝜃)
97, 8mpgbir 1873 . . 3 𝑖(𝑖𝑛 → (𝜏𝜃))
10 df-ral 3053 . . 3 (∀𝑖𝑛 (𝜏𝜃) ↔ ∀𝑖(𝑖𝑛 → (𝜏𝜃)))
119, 10mpbir 221 . 2 𝑖𝑛 (𝜏𝜃)
12 vex 3341 . . 3 𝑛 ∈ V
13 bnj1133.7 . . 3 (𝜏 ↔ ∀𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜃))
1412, 13bnj110 31233 . 2 (( E Fr 𝑛 ∧ ∀𝑖𝑛 (𝜏𝜃)) → ∀𝑖𝑛 𝜃)
154, 11, 14sylancl 697 1 (𝜒 → ∀𝑖𝑛 𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1628   = wceq 1630  wcel 2137  wral 3048  [wsbc 3574  cdif 3710  c0 4056  {csn 4319   class class class wbr 4802   E cep 5176   Fr wfr 5220   Fn wfn 6042  ωcom 7228  w-bnj17 31059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pr 5053  ax-un 7112
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-br 4803  df-opab 4863  df-tr 4903  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-we 5225  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-om 7229  df-bnj17 31060
This theorem is referenced by:  bnj1128  31363
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