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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 3444 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | 1 | frege54cor1c 40616 | . . 3 ⊢ [𝑥 / 𝑦]𝑦 = 𝑥 |
3 | frege53c 40615 | . . 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 3721 | . . . 4 ⊢ ([𝐴 / 𝑦]𝑦 = 𝑥 ↔ 𝐴 ∈ {𝑦 ∣ 𝑦 = 𝑥}) | |
6 | clelab 2933 | . . . 4 ⊢ (𝐴 ∈ {𝑦 ∣ 𝑦 = 𝑥} ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = 𝑥)) | |
7 | 5, 6 | bitri 278 | . . 3 ⊢ ([𝐴 / 𝑦]𝑦 = 𝑥 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = 𝑥)) |
8 | eqtr2 2819 | . . . 4 ⊢ ((𝑦 = 𝐴 ∧ 𝑦 = 𝑥) → 𝐴 = 𝑥) | |
9 | 8 | exlimiv 1931 | . . 3 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = 𝑥) → 𝐴 = 𝑥) |
10 | 7, 9 | sylbi 220 | . 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 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 {cab 2776 Vcvv 3441 [wsbc 3720 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-frege8 40510 ax-frege52c 40589 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-sbc 3721 df-sn 4526 |
This theorem is referenced by: frege104 40668 |
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