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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 3477 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | 1 | frege54cor1c 43131 | . . 3 ⊢ [𝑥 / 𝑦]𝑦 = 𝑥 |
3 | frege53c 43130 | . . 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 3778 | . . . 4 ⊢ ([𝐴 / 𝑦]𝑦 = 𝑥 ↔ 𝐴 ∈ {𝑦 ∣ 𝑦 = 𝑥}) | |
6 | clelab 2878 | . . . 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 2755 | . . . 4 ⊢ ((𝑦 = 𝐴 ∧ 𝑦 = 𝑥) → 𝐴 = 𝑥) | |
9 | 8 | exlimiv 1932 | . . 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 395 = wceq 1540 ∃wex 1780 ∈ wcel 2105 {cab 2708 Vcvv 3473 [wsbc 3777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-12 2170 ax-ext 2702 ax-frege8 43025 ax-frege52c 43104 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1543 df-ex 1781 df-nf 1785 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-v 3475 df-sbc 3778 df-sn 4629 |
This theorem is referenced by: frege104 43183 |
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