Description: Commutation law for identical variable specifiers. The antecedent and consequent are true when x and y are substituted with the same variable. Lemma L12 in Megill p. 445 (p. 12 of the preprint). Version of aecom using ax-c11 . Unlike axc11nfromc11 , this version does not require ax-5 (see comment of equcomi1 ). (Contributed by NM, 10-May-1993) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | aecom-o | |- ( A. x x = y -> A. y y = x ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ax-c11 | |- ( A. x x = y -> ( A. x x = y -> A. y x = y ) ) |
|
| 2 | 1 | pm2.43i | |- ( A. x x = y -> A. y x = y ) |
| 3 | equcomi1 | |- ( x = y -> y = x ) |
|
| 4 | 3 | alimi | |- ( A. y x = y -> A. y y = x ) |
| 5 | 2 4 | syl | |- ( A. x x = y -> A. y y = x ) |