Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj121 | Structured version Visualization version GIF version |
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj121.1 | ⊢ (𝜁 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))) |
bnj121.2 | ⊢ (𝜁′ ↔ [1o / 𝑛]𝜁) |
bnj121.3 | ⊢ (𝜑′ ↔ [1o / 𝑛]𝜑) |
bnj121.4 | ⊢ (𝜓′ ↔ [1o / 𝑛]𝜓) |
Ref | Expression |
---|---|
bnj121 | ⊢ (𝜁′ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj121.1 | . . 3 ⊢ (𝜁 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))) | |
2 | 1 | sbcbii 3736 | . 2 ⊢ ([1o / 𝑛]𝜁 ↔ [1o / 𝑛]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))) |
3 | bnj121.2 | . 2 ⊢ (𝜁′ ↔ [1o / 𝑛]𝜁) | |
4 | bnj105 32265 | . . . . . . . 8 ⊢ 1o ∈ V | |
5 | 4 | bnj90 32263 | . . . . . . 7 ⊢ ([1o / 𝑛]𝑓 Fn 𝑛 ↔ 𝑓 Fn 1o) |
6 | 5 | bicomi 227 | . . . . . 6 ⊢ (𝑓 Fn 1o ↔ [1o / 𝑛]𝑓 Fn 𝑛) |
7 | bnj121.3 | . . . . . 6 ⊢ (𝜑′ ↔ [1o / 𝑛]𝜑) | |
8 | bnj121.4 | . . . . . 6 ⊢ (𝜓′ ↔ [1o / 𝑛]𝜓) | |
9 | 6, 7, 8 | 3anbi123i 1156 | . . . . 5 ⊢ ((𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) ↔ ([1o / 𝑛]𝑓 Fn 𝑛 ∧ [1o / 𝑛]𝜑 ∧ [1o / 𝑛]𝜓)) |
10 | sbc3an 3745 | . . . . 5 ⊢ ([1o / 𝑛](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ([1o / 𝑛]𝑓 Fn 𝑛 ∧ [1o / 𝑛]𝜑 ∧ [1o / 𝑛]𝜓)) | |
11 | 9, 10 | bitr4i 281 | . . . 4 ⊢ ((𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) ↔ [1o / 𝑛](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) |
12 | 11 | imbi2i 339 | . . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → [1o / 𝑛](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))) |
13 | nfv 1920 | . . . . 5 ⊢ Ⅎ𝑛(𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) | |
14 | 13 | sbc19.21g 3752 | . . . 4 ⊢ (1o ∈ V → ([1o / 𝑛]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → [1o / 𝑛](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))) |
15 | 4, 14 | ax-mp 5 | . . 3 ⊢ ([1o / 𝑛]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → [1o / 𝑛](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))) |
16 | 12, 15 | bitr4i 281 | . 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) ↔ [1o / 𝑛]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))) |
17 | 2, 3, 16 | 3bitr4i 306 | 1 ⊢ (𝜁′ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1088 ∈ wcel 2113 Vcvv 3397 [wsbc 3679 Fn wfn 6328 1oc1o 8117 FrSe w-bnj15 32233 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-v 3399 df-sbc 3680 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-pw 4487 df-sn 4514 df-suc 6172 df-fn 6336 df-1o 8124 |
This theorem is referenced by: bnj150 32419 bnj153 32423 |
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