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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1133 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 32277. 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.) |
Ref | Expression |
---|---|
bnj1133.3 | ⊢ 𝐷 = (ω ∖ {∅}) |
bnj1133.5 | ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) |
bnj1133.7 | ⊢ (𝜏 ↔ ∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜃)) |
bnj1133.8 | ⊢ ((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃) |
Ref | Expression |
---|---|
bnj1133 | ⊢ (𝜒 → ∀𝑖 ∈ 𝑛 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1133.5 | . . 3 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) | |
2 | bnj1133.3 | . . . 4 ⊢ 𝐷 = (ω ∖ {∅}) | |
3 | 2 | bnj1071 32244 | . . 3 ⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛) |
4 | 1, 3 | bnj769 32028 | . 2 ⊢ (𝜒 → E Fr 𝑛) |
5 | impexp 453 | . . . . . 6 ⊢ (((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃) ↔ (𝑖 ∈ 𝑛 → (𝜏 → 𝜃))) | |
6 | 5 | bicomi 226 | . . . . 5 ⊢ ((𝑖 ∈ 𝑛 → (𝜏 → 𝜃)) ↔ ((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃)) |
7 | 6 | albii 1816 | . . . 4 ⊢ (∀𝑖(𝑖 ∈ 𝑛 → (𝜏 → 𝜃)) ↔ ∀𝑖((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃)) |
8 | bnj1133.8 | . . . 4 ⊢ ((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃) | |
9 | 7, 8 | mpgbir 1796 | . . 3 ⊢ ∀𝑖(𝑖 ∈ 𝑛 → (𝜏 → 𝜃)) |
10 | df-ral 3143 | . . 3 ⊢ (∀𝑖 ∈ 𝑛 (𝜏 → 𝜃) ↔ ∀𝑖(𝑖 ∈ 𝑛 → (𝜏 → 𝜃))) | |
11 | 9, 10 | mpbir 233 | . 2 ⊢ ∀𝑖 ∈ 𝑛 (𝜏 → 𝜃) |
12 | vex 3498 | . . 3 ⊢ 𝑛 ∈ V | |
13 | bnj1133.7 | . . 3 ⊢ (𝜏 ↔ ∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜃)) | |
14 | 12, 13 | bnj110 32125 | . 2 ⊢ (( E Fr 𝑛 ∧ ∀𝑖 ∈ 𝑛 (𝜏 → 𝜃)) → ∀𝑖 ∈ 𝑛 𝜃) |
15 | 4, 11, 14 | sylancl 588 | 1 ⊢ (𝜒 → ∀𝑖 ∈ 𝑛 𝜃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1531 = wceq 1533 ∈ wcel 2110 ∀wral 3138 [wsbc 3772 ∖ cdif 3933 ∅c0 4291 {csn 4561 class class class wbr 5059 E cep 5459 Fr wfr 5506 Fn wfn 6345 ωcom 7574 ∧ w-bnj17 31951 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-tr 5166 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-om 7575 df-bnj17 31952 |
This theorem is referenced by: bnj1128 32257 |
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