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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege104 | Structured version Visualization version GIF version |
Description: Proposition 104 of [Frege1879] p. 73.
Note: in the Bauer-Meenfelberg translation published in van Heijenoort's collection From Frege to Goedel, this proof has the minor clause and result swapped. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege103.z | ⊢ 𝑍 ∈ 𝑉 |
Ref | Expression |
---|---|
frege104 | ⊢ (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍 → 𝑋 = 𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege103.z | . . . 4 ⊢ 𝑍 ∈ 𝑉 | |
2 | 1 | elexi 3511 | . . 3 ⊢ 𝑍 ∈ V |
3 | eqeq1 2822 | . . . 4 ⊢ (𝑧 = 𝑍 → (𝑧 = 𝑋 ↔ 𝑍 = 𝑋)) | |
4 | eqeq2 2830 | . . . 4 ⊢ (𝑧 = 𝑍 → (𝑋 = 𝑧 ↔ 𝑋 = 𝑍)) | |
5 | 3, 4 | imbi12d 346 | . . 3 ⊢ (𝑧 = 𝑍 → ((𝑧 = 𝑋 → 𝑋 = 𝑧) ↔ (𝑍 = 𝑋 → 𝑋 = 𝑍))) |
6 | frege55c 40142 | . . 3 ⊢ (𝑧 = 𝑋 → 𝑋 = 𝑧) | |
7 | 2, 5, 6 | vtocl 3557 | . 2 ⊢ (𝑍 = 𝑋 → 𝑋 = 𝑍) |
8 | 1 | frege103 40190 | . 2 ⊢ ((𝑍 = 𝑋 → 𝑋 = 𝑍) → (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍 → 𝑋 = 𝑍))) |
9 | 7, 8 | ax-mp 5 | 1 ⊢ (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍 → 𝑋 = 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1528 ∈ wcel 2105 ∪ cun 3931 class class class wbr 5057 I cid 5452 ‘cfv 6348 t+ctcl 14333 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-frege1 40014 ax-frege2 40015 ax-frege8 40033 ax-frege52a 40081 ax-frege52c 40112 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ifp 1055 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 |
This theorem is referenced by: frege114 40201 |
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