| Mathbox for Jonathan Ben-Naim |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1133 | Structured version Visualization version GIF version | ||
| Description: Technical lemma for bnj69 35024. 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 34991 | . . 3 ⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛) |
| 4 | 1, 3 | bnj769 34776 | . 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 1819 | . . . 4 ⊢ (∀𝑖(𝑖 ∈ 𝑛 → (𝜏 → 𝜃)) ↔ ∀𝑖((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃)) |
| 8 | bnj1133.8 | . . . 4 ⊢ ((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃) | |
| 9 | 7, 8 | mpgbir 1799 | . . 3 ⊢ ∀𝑖(𝑖 ∈ 𝑛 → (𝜏 → 𝜃)) |
| 10 | df-ral 3062 | . . 3 ⊢ (∀𝑖 ∈ 𝑛 (𝜏 → 𝜃) ↔ ∀𝑖(𝑖 ∈ 𝑛 → (𝜏 → 𝜃))) | |
| 11 | 9, 10 | mpbir 231 | . 2 ⊢ ∀𝑖 ∈ 𝑛 (𝜏 → 𝜃) |
| 12 | vex 3484 | . . 3 ⊢ 𝑛 ∈ V | |
| 13 | bnj1133.7 | . . 3 ⊢ (𝜏 ↔ ∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜃)) | |
| 14 | 12, 13 | bnj110 34872 | . 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 1538 = wceq 1540 ∈ wcel 2108 ∀wral 3061 [wsbc 3788 ∖ cdif 3948 ∅c0 4333 {csn 4626 class class class wbr 5143 E cep 5583 Fr wfr 5634 Fn wfn 6556 ωcom 7887 ∧ w-bnj17 34700 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-tr 5260 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 df-om 7888 df-bnj17 34701 |
| This theorem is referenced by: bnj1128 35004 |
| Copyright terms: Public domain | W3C validator |