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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1133 | Structured version Visualization version GIF version |
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.) |
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 33646 | . . 3 ⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛) |
4 | 1, 3 | bnj769 33431 | . 2 ⊢ (𝜒 → E Fr 𝑛) |
5 | impexp 452 | . . . . . 6 ⊢ (((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃) ↔ (𝑖 ∈ 𝑛 → (𝜏 → 𝜃))) | |
6 | 5 | bicomi 223 | . . . . 5 ⊢ ((𝑖 ∈ 𝑛 → (𝜏 → 𝜃)) ↔ ((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃)) |
7 | 6 | albii 1822 | . . . 4 ⊢ (∀𝑖(𝑖 ∈ 𝑛 → (𝜏 → 𝜃)) ↔ ∀𝑖((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃)) |
8 | bnj1133.8 | . . . 4 ⊢ ((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃) | |
9 | 7, 8 | mpgbir 1802 | . . 3 ⊢ ∀𝑖(𝑖 ∈ 𝑛 → (𝜏 → 𝜃)) |
10 | df-ral 3062 | . . 3 ⊢ (∀𝑖 ∈ 𝑛 (𝜏 → 𝜃) ↔ ∀𝑖(𝑖 ∈ 𝑛 → (𝜏 → 𝜃))) | |
11 | 9, 10 | mpbir 230 | . 2 ⊢ ∀𝑖 ∈ 𝑛 (𝜏 → 𝜃) |
12 | vex 3448 | . . 3 ⊢ 𝑛 ∈ V | |
13 | bnj1133.7 | . . 3 ⊢ (𝜏 ↔ ∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜃)) | |
14 | 12, 13 | bnj110 33527 | . 2 ⊢ (( E Fr 𝑛 ∧ ∀𝑖 ∈ 𝑛 (𝜏 → 𝜃)) → ∀𝑖 ∈ 𝑛 𝜃) |
15 | 4, 11, 14 | sylancl 587 | 1 ⊢ (𝜒 → ∀𝑖 ∈ 𝑛 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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