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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege113 | Structured version Visualization version GIF version |
Description: Proposition 113 of [Frege1879] p. 76. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege112.z | ⊢ 𝑍 ∈ 𝑉 |
Ref | Expression |
---|---|
frege113 | ⊢ ((𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋 → 𝑍 = 𝑋)) → (𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋 → 𝑋((t+‘𝑅) ∪ I )𝑍))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege112.z | . . 3 ⊢ 𝑍 ∈ 𝑉 | |
2 | 1 | frege112 39109 | . 2 ⊢ (𝑍 = 𝑋 → 𝑋((t+‘𝑅) ∪ I )𝑍) |
3 | frege7 38942 | . 2 ⊢ ((𝑍 = 𝑋 → 𝑋((t+‘𝑅) ∪ I )𝑍) → ((𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋 → 𝑍 = 𝑋)) → (𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋 → 𝑋((t+‘𝑅) ∪ I )𝑍)))) | |
4 | 2, 3 | ax-mp 5 | 1 ⊢ ((𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋 → 𝑍 = 𝑋)) → (𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋 → 𝑋((t+‘𝑅) ∪ I )𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1658 ∈ wcel 2166 ∪ cun 3796 class class class wbr 4873 I cid 5249 ‘cfv 6123 t+ctcl 14103 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 ax-frege1 38924 ax-frege2 38925 ax-frege8 38943 ax-frege52a 38991 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-ifp 1092 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-br 4874 df-opab 4936 df-id 5250 df-xp 5348 df-rel 5349 |
This theorem is referenced by: frege114 39111 |
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