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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege55c | Structured version Visualization version GIF version |
Description: Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege55c | ⊢ (𝑥 = 𝐴 → 𝐴 = 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3452 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | 1 | frege54cor1c 42261 | . . 3 ⊢ [𝑥 / 𝑦]𝑦 = 𝑥 |
3 | frege53c 42260 | . . 3 ⊢ ([𝑥 / 𝑦]𝑦 = 𝑥 → (𝑥 = 𝐴 → [𝐴 / 𝑦]𝑦 = 𝑥)) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (𝑥 = 𝐴 → [𝐴 / 𝑦]𝑦 = 𝑥) |
5 | df-sbc 3745 | . . . 4 ⊢ ([𝐴 / 𝑦]𝑦 = 𝑥 ↔ 𝐴 ∈ {𝑦 ∣ 𝑦 = 𝑥}) | |
6 | clelab 2884 | . . . 4 ⊢ (𝐴 ∈ {𝑦 ∣ 𝑦 = 𝑥} ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = 𝑥)) | |
7 | 5, 6 | bitri 275 | . . 3 ⊢ ([𝐴 / 𝑦]𝑦 = 𝑥 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = 𝑥)) |
8 | eqtr2 2761 | . . . 4 ⊢ ((𝑦 = 𝐴 ∧ 𝑦 = 𝑥) → 𝐴 = 𝑥) | |
9 | 8 | exlimiv 1934 | . . 3 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = 𝑥) → 𝐴 = 𝑥) |
10 | 7, 9 | sylbi 216 | . 2 ⊢ ([𝐴 / 𝑦]𝑦 = 𝑥 → 𝐴 = 𝑥) |
11 | 4, 10 | syl 17 | 1 ⊢ (𝑥 = 𝐴 → 𝐴 = 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 {cab 2714 Vcvv 3448 [wsbc 3744 |
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-10 2138 ax-12 2172 ax-ext 2708 ax-frege8 42155 ax-frege52c 42234 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-v 3450 df-sbc 3745 df-sn 4592 |
This theorem is referenced by: frege104 42313 |
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