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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 35003. 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 34970 | . . 3 ⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛) |
4 | 1, 3 | bnj769 34755 | . 2 ⊢ (𝜒 → E Fr 𝑛) |
5 | impexp 450 | . . . . . 6 ⊢ (((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃) ↔ (𝑖 ∈ 𝑛 → (𝜏 → 𝜃))) | |
6 | 5 | bicomi 224 | . . . . 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 3060 | . . 3 ⊢ (∀𝑖 ∈ 𝑛 (𝜏 → 𝜃) ↔ ∀𝑖(𝑖 ∈ 𝑛 → (𝜏 → 𝜃))) | |
11 | 9, 10 | mpbir 231 | . 2 ⊢ ∀𝑖 ∈ 𝑛 (𝜏 → 𝜃) |
12 | vex 3482 | . . 3 ⊢ 𝑛 ∈ V | |
13 | bnj1133.7 | . . 3 ⊢ (𝜏 ↔ ∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜃)) | |
14 | 12, 13 | bnj110 34851 | . 2 ⊢ (( E Fr 𝑛 ∧ ∀𝑖 ∈ 𝑛 (𝜏 → 𝜃)) → ∀𝑖 ∈ 𝑛 𝜃) |
15 | 4, 11, 14 | sylancl 586 | 1 ⊢ (𝜒 → ∀𝑖 ∈ 𝑛 𝜃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1535 = wceq 1537 ∈ wcel 2106 ∀wral 3059 [wsbc 3791 ∖ cdif 3960 ∅c0 4339 {csn 4631 class class class wbr 5148 E cep 5588 Fr wfr 5638 Fn wfn 6558 ωcom 7887 ∧ w-bnj17 34679 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-tr 5266 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-ord 6389 df-on 6390 df-om 7888 df-bnj17 34680 |
This theorem is referenced by: bnj1128 34983 |
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