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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege107 | Structured version Visualization version GIF version |
Description: Proposition 107 of [Frege1879] p. 74. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege107.v | ⊢ 𝑉 ∈ 𝐴 |
Ref | Expression |
---|---|
frege107 | ⊢ ((𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉 → 𝑍(t+‘𝑅)𝑉)) → (𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉 → 𝑍((t+‘𝑅) ∪ I )𝑉))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege107.v | . . 3 ⊢ 𝑉 ∈ 𝐴 | |
2 | 1 | frege106 43430 | . 2 ⊢ (𝑍(t+‘𝑅)𝑉 → 𝑍((t+‘𝑅) ∪ I )𝑉) |
3 | frege7 43269 | . 2 ⊢ ((𝑍(t+‘𝑅)𝑉 → 𝑍((t+‘𝑅) ∪ I )𝑉) → ((𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉 → 𝑍(t+‘𝑅)𝑉)) → (𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉 → 𝑍((t+‘𝑅) ∪ I )𝑉)))) | |
4 | 2, 3 | ax-mp 5 | 1 ⊢ ((𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉 → 𝑍(t+‘𝑅)𝑉)) → (𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉 → 𝑍((t+‘𝑅) ∪ I )𝑉))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 ∪ cun 3947 class class class wbr 5152 I cid 5579 ‘cfv 6553 t+ctcl 14972 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-frege1 43251 ax-frege2 43252 ax-frege8 43270 ax-frege28 43291 ax-frege31 43295 ax-frege52a 43318 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ifp 1061 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-br 5153 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 |
This theorem is referenced by: frege108 43432 |
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