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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege72 | Structured version Visualization version GIF version |
Description: If property 𝐴 is hereditary in the 𝑅-sequence, if 𝑥 has property 𝐴, and if 𝑦 is a result of an application of the procedure 𝑅 to 𝑥, then 𝑦 has property 𝐴. Proposition 72 of [Frege1879] p. 59. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege72.x | ⊢ 𝑋 ∈ 𝑈 |
frege72.y | ⊢ 𝑌 ∈ 𝑉 |
Ref | Expression |
---|---|
frege72 | ⊢ (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege72.y | . . . 4 ⊢ 𝑌 ∈ 𝑉 | |
2 | 1 | frege58c 43264 | . . 3 ⊢ (∀𝑧(𝑋𝑅𝑧 → 𝑧 ∈ 𝐴) → [𝑌 / 𝑧](𝑋𝑅𝑧 → 𝑧 ∈ 𝐴)) |
3 | sbcim1 3830 | . . . 4 ⊢ ([𝑌 / 𝑧](𝑋𝑅𝑧 → 𝑧 ∈ 𝐴) → ([𝑌 / 𝑧]𝑋𝑅𝑧 → [𝑌 / 𝑧]𝑧 ∈ 𝐴)) | |
4 | sbcbr2g 5200 | . . . . . 6 ⊢ (𝑌 ∈ 𝑉 → ([𝑌 / 𝑧]𝑋𝑅𝑧 ↔ 𝑋𝑅⦋𝑌 / 𝑧⦌𝑧)) | |
5 | csbvarg 4427 | . . . . . . 7 ⊢ (𝑌 ∈ 𝑉 → ⦋𝑌 / 𝑧⦌𝑧 = 𝑌) | |
6 | 5 | breq2d 5154 | . . . . . 6 ⊢ (𝑌 ∈ 𝑉 → (𝑋𝑅⦋𝑌 / 𝑧⦌𝑧 ↔ 𝑋𝑅𝑌)) |
7 | 4, 6 | bitrd 279 | . . . . 5 ⊢ (𝑌 ∈ 𝑉 → ([𝑌 / 𝑧]𝑋𝑅𝑧 ↔ 𝑋𝑅𝑌)) |
8 | 1, 7 | ax-mp 5 | . . . 4 ⊢ ([𝑌 / 𝑧]𝑋𝑅𝑧 ↔ 𝑋𝑅𝑌) |
9 | sbcel1v 3844 | . . . 4 ⊢ ([𝑌 / 𝑧]𝑧 ∈ 𝐴 ↔ 𝑌 ∈ 𝐴) | |
10 | 3, 8, 9 | 3imtr3g 295 | . . 3 ⊢ ([𝑌 / 𝑧](𝑋𝑅𝑧 → 𝑧 ∈ 𝐴) → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴)) |
11 | 2, 10 | syl 17 | . 2 ⊢ (∀𝑧(𝑋𝑅𝑧 → 𝑧 ∈ 𝐴) → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴)) |
12 | frege72.x | . . 3 ⊢ 𝑋 ∈ 𝑈 | |
13 | 12 | frege71 43277 | . 2 ⊢ ((∀𝑧(𝑋𝑅𝑧 → 𝑧 ∈ 𝐴) → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴)) → (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴)))) |
14 | 11, 13 | ax-mp 5 | 1 ⊢ (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1532 ∈ wcel 2099 [wsbc 3774 ⦋csb 3889 class class class wbr 5142 hereditary whe 43115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-frege1 43133 ax-frege2 43134 ax-frege8 43152 ax-frege52a 43200 ax-frege58b 43244 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-ifp 1062 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 df-opab 5205 df-xp 5678 df-cnv 5680 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-he 43116 |
This theorem is referenced by: frege73 43279 frege74 43280 |
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