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 3458 | . 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 30044 | . . . . . . . 8 ⊢ 1𝑜 ∈ V | |
5 | 4 | bnj90 30042 | . . . . . . 7 ⊢ ([1𝑜 / 𝑛]𝑓 Fn 𝑛 ↔ 𝑓 Fn 1𝑜) |
6 | 5 | bicomi 213 | . . . . . 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 1244 | . . . . 5 ⊢ ((𝑓 Fn 1𝑜 ∧ 𝜑′ ∧ 𝜓′) ↔ ([1𝑜 / 𝑛]𝑓 Fn 𝑛 ∧ [1𝑜 / 𝑛]𝜑 ∧ [1𝑜 / 𝑛]𝜓)) |
10 | sbc3an 3461 | . . . . 5 ⊢ ([1𝑜 / 𝑛](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ([1𝑜 / 𝑛]𝑓 Fn 𝑛 ∧ [1𝑜 / 𝑛]𝜑 ∧ [1𝑜 / 𝑛]𝜓)) | |
11 | 9, 10 | bitr4i 266 | . . . 4 ⊢ ((𝑓 Fn 1𝑜 ∧ 𝜑′ ∧ 𝜓′) ↔ [1𝑜 / 𝑛](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) |
12 | 11 | imbi2i 325 | . . 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 1830 | . . . . 5 ⊢ Ⅎ𝑛(𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) | |
14 | 13 | sbc19.21g 3469 | . . . 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 266 | . 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 291 | 1 ⊢ (𝜁′ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1𝑜 ∧ 𝜑′ ∧ 𝜓′))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 ∈ wcel 1977 Vcvv 3173 [wsbc 3402 Fn wfn 5799 1𝑜c1o 7440 FrSe w-bnj15 30011 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-pw 4110 df-sn 4126 df-suc 5646 df-fn 5807 df-1o 7447 |
This theorem is referenced by: bnj150 30200 bnj153 30204 |
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