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Mirrors > Home > ILE Home > Th. List > equid | GIF version |
Description: Identity law for equality
(reflexivity). Lemma 6 of [Tarski] p. 68.
This is often an axiom of equality in textbook systems, but we don't
need it as an axiom since it can be proved from our other axioms.
This proof is similar to Tarski's and makes use of a dummy variable 𝑦. It also works in intuitionistic logic, unlike some other possible ways of proving this theorem. (Contributed by NM, 1-Apr-2005.) |
Ref | Expression |
---|---|
equid | ⊢ 𝑥 = 𝑥 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | a9e 1659 | . 2 ⊢ ∃𝑦 𝑦 = 𝑥 | |
2 | ax-17 1491 | . . 3 ⊢ (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥) | |
3 | ax-8 1467 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝑥 → 𝑥 = 𝑥)) | |
4 | 3 | pm2.43i 49 | . . 3 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑥) |
5 | 2, 4 | exlimih 1557 | . 2 ⊢ (∃𝑦 𝑦 = 𝑥 → 𝑥 = 𝑥) |
6 | 1, 5 | ax-mp 5 | 1 ⊢ 𝑥 = 𝑥 |
Colors of variables: wff set class |
Syntax hints: ∃wex 1453 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-gen 1410 ax-ie2 1455 ax-8 1467 ax-17 1491 ax-i9 1495 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: nfequid 1663 stdpc6 1664 equcomi 1665 equveli 1717 sbid 1732 ax16i 1814 exists1 2073 vjust 2661 vex 2663 reu6 2846 nfccdeq 2880 sbc8g 2889 dfnul3 3336 rab0 3361 int0 3755 ruv 4435 dcextest 4465 relop 4659 f1eqcocnv 5660 mpoxopoveq 6105 snexxph 6806 |
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