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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 3515 | . . 3 ⊢ 𝑍 ∈ V |
3 | eqeq1 2827 | . . . 4 ⊢ (𝑧 = 𝑍 → (𝑧 = 𝑋 ↔ 𝑍 = 𝑋)) | |
4 | eqeq2 2835 | . . . 4 ⊢ (𝑧 = 𝑍 → (𝑋 = 𝑧 ↔ 𝑋 = 𝑍)) | |
5 | 3, 4 | imbi12d 347 | . . 3 ⊢ (𝑧 = 𝑍 → ((𝑧 = 𝑋 → 𝑋 = 𝑧) ↔ (𝑍 = 𝑋 → 𝑋 = 𝑍))) |
6 | frege55c 40271 | . . 3 ⊢ (𝑧 = 𝑋 → 𝑋 = 𝑧) | |
7 | 2, 5, 6 | vtocl 3561 | . 2 ⊢ (𝑍 = 𝑋 → 𝑋 = 𝑍) |
8 | 1 | frege103 40319 | . 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 1537 ∈ wcel 2114 ∪ cun 3936 class class class wbr 5068 I cid 5461 ‘cfv 6357 t+ctcl 14347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-frege1 40143 ax-frege2 40144 ax-frege8 40162 ax-frege52a 40210 ax-frege52c 40241 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 |
This theorem is referenced by: frege114 40330 |
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