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 3491 | . 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 30790 | . . . . . . . 8 ⊢ 1𝑜 ∈ V | |
5 | 4 | bnj90 30788 | . . . . . . 7 ⊢ ([1𝑜 / 𝑛]𝑓 Fn 𝑛 ↔ 𝑓 Fn 1𝑜) |
6 | 5 | bicomi 214 | . . . . . 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 1251 | . . . . 5 ⊢ ((𝑓 Fn 1𝑜 ∧ 𝜑′ ∧ 𝜓′) ↔ ([1𝑜 / 𝑛]𝑓 Fn 𝑛 ∧ [1𝑜 / 𝑛]𝜑 ∧ [1𝑜 / 𝑛]𝜓)) |
10 | sbc3an 3494 | . . . . 5 ⊢ ([1𝑜 / 𝑛](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ([1𝑜 / 𝑛]𝑓 Fn 𝑛 ∧ [1𝑜 / 𝑛]𝜑 ∧ [1𝑜 / 𝑛]𝜓)) | |
11 | 9, 10 | bitr4i 267 | . . . 4 ⊢ ((𝑓 Fn 1𝑜 ∧ 𝜑′ ∧ 𝜓′) ↔ [1𝑜 / 𝑛](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) |
12 | 11 | imbi2i 326 | . . 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 1843 | . . . . 5 ⊢ Ⅎ𝑛(𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) | |
14 | 13 | sbc19.21g 3502 | . . . 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 267 | . 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 292 | 1 ⊢ (𝜁′ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1𝑜 ∧ 𝜑′ ∧ 𝜓′))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 ∈ wcel 1990 Vcvv 3200 [wsbc 3435 Fn wfn 5883 1𝑜c1o 7553 FrSe w-bnj15 30758 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-pw 4160 df-sn 4178 df-suc 5729 df-fn 5891 df-1o 7560 |
This theorem is referenced by: bnj150 30946 bnj153 30950 |
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