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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 42653 | . 2 ⊢ (𝑍(t+‘𝑅)𝑉 → 𝑍((t+‘𝑅) ∪ I )𝑉) |
3 | frege7 42492 | . 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 2107 ∪ cun 3945 class class class wbr 5147 I cid 5572 ‘cfv 6540 t+ctcl 14928 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-frege1 42474 ax-frege2 42475 ax-frege8 42493 ax-frege28 42514 ax-frege31 42518 ax-frege52a 42541 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ifp 1063 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 |
This theorem is referenced by: frege108 42655 |
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