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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 1583 | . 2 ⊢ ∃y y = x | |
2 | ax-17 1416 | . . 3 ⊢ (x = x → ∀y x = x) | |
3 | ax-8 1392 | . . . 4 ⊢ (y = x → (y = x → x = x)) | |
4 | 3 | pm2.43i 43 | . . 3 ⊢ (y = x → x = x) |
5 | 2, 4 | exlimih 1481 | . 2 ⊢ (∃y y = x → x = x) |
6 | 1, 5 | ax-mp 7 | 1 ⊢ x = x |
Colors of variables: wff set class |
Syntax hints: ∃wex 1378 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-gen 1335 ax-ie2 1380 ax-8 1392 ax-17 1416 ax-i9 1420 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: nfequid 1587 stdpc6 1588 equcomi 1589 equveli 1639 sbid 1654 ax16i 1735 exists1 1993 vjust 2552 vex 2554 reu6 2724 nfccdeq 2756 sbc8g 2765 dfnul3 3221 rab0 3240 int0 3620 ruv 4228 relop 4429 f1eqcocnv 5374 mpt2xopoveq 5796 |
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