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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege70 | Structured version Visualization version GIF version |
Description: Lemma for frege72 43175. Proposition 70 of [Frege1879] p. 58. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 3-Jul-2020.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege70.x | ⊢ 𝑋 ∈ 𝑉 |
Ref | Expression |
---|---|
frege70 | ⊢ (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → ∀𝑦(𝑋𝑅𝑦 → 𝑦 ∈ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffrege69 43172 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ 𝑅 hereditary 𝐴) | |
2 | frege70.x | . . . 4 ⊢ 𝑋 ∈ 𝑉 | |
3 | 2 | frege68c 43171 | . . 3 ⊢ ((∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ 𝑅 hereditary 𝐴) → (𝑅 hereditary 𝐴 → [𝑋 / 𝑥](𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)))) |
4 | sbcel1v 3840 | . . . . 5 ⊢ ([𝑋 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴) | |
5 | 4 | biimpri 227 | . . . 4 ⊢ (𝑋 ∈ 𝐴 → [𝑋 / 𝑥]𝑥 ∈ 𝐴) |
6 | sbcim1 3825 | . . . 4 ⊢ ([𝑋 / 𝑥](𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) → ([𝑋 / 𝑥]𝑥 ∈ 𝐴 → [𝑋 / 𝑥]∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴))) | |
7 | sbcal 3833 | . . . . 5 ⊢ ([𝑋 / 𝑥]∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴) ↔ ∀𝑦[𝑋 / 𝑥](𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) | |
8 | sbcim1 3825 | . . . . . . 7 ⊢ ([𝑋 / 𝑥](𝑥𝑅𝑦 → 𝑦 ∈ 𝐴) → ([𝑋 / 𝑥]𝑥𝑅𝑦 → [𝑋 / 𝑥]𝑦 ∈ 𝐴)) | |
9 | sbcbr1g 5195 | . . . . . . . . 9 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]𝑥𝑅𝑦 ↔ ⦋𝑋 / 𝑥⦌𝑥𝑅𝑦)) | |
10 | 2, 9 | ax-mp 5 | . . . . . . . 8 ⊢ ([𝑋 / 𝑥]𝑥𝑅𝑦 ↔ ⦋𝑋 / 𝑥⦌𝑥𝑅𝑦) |
11 | csbvarg 4423 | . . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑥⦌𝑥 = 𝑋) | |
12 | 2, 11 | ax-mp 5 | . . . . . . . . 9 ⊢ ⦋𝑋 / 𝑥⦌𝑥 = 𝑋 |
13 | 12 | breq1i 5145 | . . . . . . . 8 ⊢ (⦋𝑋 / 𝑥⦌𝑥𝑅𝑦 ↔ 𝑋𝑅𝑦) |
14 | 10, 13 | bitri 275 | . . . . . . 7 ⊢ ([𝑋 / 𝑥]𝑥𝑅𝑦 ↔ 𝑋𝑅𝑦) |
15 | sbcg 3848 | . . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
16 | 2, 15 | ax-mp 5 | . . . . . . 7 ⊢ ([𝑋 / 𝑥]𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) |
17 | 8, 14, 16 | 3imtr3g 295 | . . . . . 6 ⊢ ([𝑋 / 𝑥](𝑥𝑅𝑦 → 𝑦 ∈ 𝐴) → (𝑋𝑅𝑦 → 𝑦 ∈ 𝐴)) |
18 | 17 | alimi 1805 | . . . . 5 ⊢ (∀𝑦[𝑋 / 𝑥](𝑥𝑅𝑦 → 𝑦 ∈ 𝐴) → ∀𝑦(𝑋𝑅𝑦 → 𝑦 ∈ 𝐴)) |
19 | 7, 18 | sylbi 216 | . . . 4 ⊢ ([𝑋 / 𝑥]∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴) → ∀𝑦(𝑋𝑅𝑦 → 𝑦 ∈ 𝐴)) |
20 | 5, 6, 19 | syl56 36 | . . 3 ⊢ ([𝑋 / 𝑥](𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) → (𝑋 ∈ 𝐴 → ∀𝑦(𝑋𝑅𝑦 → 𝑦 ∈ 𝐴))) |
21 | 3, 20 | syl6 35 | . 2 ⊢ ((∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ 𝑅 hereditary 𝐴) → (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → ∀𝑦(𝑋𝑅𝑦 → 𝑦 ∈ 𝐴)))) |
22 | 1, 21 | ax-mp 5 | 1 ⊢ (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → ∀𝑦(𝑋𝑅𝑦 → 𝑦 ∈ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1531 = wceq 1533 ∈ wcel 2098 [wsbc 3769 ⦋csb 3885 class class class wbr 5138 hereditary whe 43012 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 ax-frege1 43030 ax-frege2 43031 ax-frege8 43049 ax-frege52a 43097 ax-frege58b 43141 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-ifp 1060 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-br 5139 df-opab 5201 df-xp 5672 df-cnv 5674 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-he 43013 |
This theorem is referenced by: frege71 43174 |
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