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| Description: Lemma for frege102 43983. 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 43981 | . 2 ⊢ (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋)) | 
| 3 | frege48 43870 | . 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 1539 ∈ wcel 2107 ∪ cun 3948 class class class wbr 5142 I cid 5576 ‘cfv 6560 t+ctcl 15025 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-frege1 43808 ax-frege2 43809 ax-frege8 43827 ax-frege28 43848 ax-frege31 43852 ax-frege41 43863 ax-frege52a 43875 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 | 
| This theorem is referenced by: frege102 43983 | 
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