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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege101 | Structured version Visualization version GIF version |
Description: Lemma for frege102 40304. Proposition 101 of [Frege1879] p. 72. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege99.z | ⊢ 𝑍 ∈ 𝑈 |
Ref | Expression |
---|---|
frege101 | ⊢ ((𝑍 = 𝑋 → (𝑍𝑅𝑉 → 𝑋(t+‘𝑅)𝑉)) → ((𝑋(t+‘𝑅)𝑍 → (𝑍𝑅𝑉 → 𝑋(t+‘𝑅)𝑉)) → (𝑋((t+‘𝑅) ∪ I )𝑍 → (𝑍𝑅𝑉 → 𝑋(t+‘𝑅)𝑉)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege99.z | . . 3 ⊢ 𝑍 ∈ 𝑈 | |
2 | 1 | frege100 40302 | . 2 ⊢ (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋)) |
3 | frege48 40191 | . 2 ⊢ ((𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋)) → ((𝑍 = 𝑋 → (𝑍𝑅𝑉 → 𝑋(t+‘𝑅)𝑉)) → ((𝑋(t+‘𝑅)𝑍 → (𝑍𝑅𝑉 → 𝑋(t+‘𝑅)𝑉)) → (𝑋((t+‘𝑅) ∪ I )𝑍 → (𝑍𝑅𝑉 → 𝑋(t+‘𝑅)𝑉))))) | |
4 | 2, 3 | ax-mp 5 | 1 ⊢ ((𝑍 = 𝑋 → (𝑍𝑅𝑉 → 𝑋(t+‘𝑅)𝑉)) → ((𝑋(t+‘𝑅)𝑍 → (𝑍𝑅𝑉 → 𝑋(t+‘𝑅)𝑉)) → (𝑋((t+‘𝑅) ∪ I )𝑍 → (𝑍𝑅𝑉 → 𝑋(t+‘𝑅)𝑉)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∈ wcel 2110 ∪ cun 3933 class class class wbr 5058 I cid 5453 ‘cfv 6349 t+ctcl 14339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 ax-frege1 40129 ax-frege2 40130 ax-frege8 40148 ax-frege28 40169 ax-frege31 40173 ax-frege41 40184 ax-frege52a 40196 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 |
This theorem is referenced by: frege102 40304 |
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