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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege104 | Structured version Visualization version GIF version |
Description: Proposition 104 of [Frege1879] p. 73.
Note: in the Bauer-Meenfelberg translation published in van Heijenoort's collection From Frege to Goedel, this proof has the minor clause and result swapped. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege103.z | ⊢ 𝑍 ∈ 𝑉 |
Ref | Expression |
---|---|
frege104 | ⊢ (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍 → 𝑋 = 𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege103.z | . . . 4 ⊢ 𝑍 ∈ 𝑉 | |
2 | 1 | elexi 3464 | . . 3 ⊢ 𝑍 ∈ V |
3 | eqeq1 2740 | . . . 4 ⊢ (𝑧 = 𝑍 → (𝑧 = 𝑋 ↔ 𝑍 = 𝑋)) | |
4 | eqeq2 2748 | . . . 4 ⊢ (𝑧 = 𝑍 → (𝑋 = 𝑧 ↔ 𝑋 = 𝑍)) | |
5 | 3, 4 | imbi12d 344 | . . 3 ⊢ (𝑧 = 𝑍 → ((𝑧 = 𝑋 → 𝑋 = 𝑧) ↔ (𝑍 = 𝑋 → 𝑋 = 𝑍))) |
6 | frege55c 42180 | . . 3 ⊢ (𝑧 = 𝑋 → 𝑋 = 𝑧) | |
7 | 2, 5, 6 | vtocl 3518 | . 2 ⊢ (𝑍 = 𝑋 → 𝑋 = 𝑍) |
8 | 1 | frege103 42228 | . 2 ⊢ ((𝑍 = 𝑋 → 𝑋 = 𝑍) → (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍 → 𝑋 = 𝑍))) |
9 | 7, 8 | ax-mp 5 | 1 ⊢ (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍 → 𝑋 = 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1541 ∈ wcel 2106 ∪ cun 3908 class class class wbr 5105 I cid 5530 ‘cfv 6496 t+ctcl 14870 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 ax-frege1 42052 ax-frege2 42053 ax-frege8 42071 ax-frege52a 42119 ax-frege52c 42150 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-ifp 1062 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-sbc 3740 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-br 5106 df-opab 5168 df-id 5531 df-xp 5639 df-rel 5640 |
This theorem is referenced by: frege114 42239 |
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