Description: Weak version of alcom . Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017) (Proof shortened by Wolf Lammen, 28-Dec-2023)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | alcomiw.1 | |- ( y = z -> ( ph <-> ps ) ) |
|
| Assertion | alcomiw | |- ( A. x A. y ph -> A. y A. x ph ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | alcomiw.1 | |- ( y = z -> ( ph <-> ps ) ) |
|
| 2 | 1 | cbvalvw | |- ( A. y ph <-> A. z ps ) |
| 3 | 2 | biimpi | |- ( A. y ph -> A. z ps ) |
| 4 | 3 | alimi | |- ( A. x A. y ph -> A. x A. z ps ) |
| 5 | ax-5 | |- ( A. x A. z ps -> A. y A. x A. z ps ) |
|
| 6 | 1 | biimprd | |- ( y = z -> ( ps -> ph ) ) |
| 7 | 6 | equcoms | |- ( z = y -> ( ps -> ph ) ) |
| 8 | 7 | spimvw | |- ( A. z ps -> ph ) |
| 9 | 8 | 2alimi | |- ( A. y A. x A. z ps -> A. y A. x ph ) |
| 10 | 4 5 9 | 3syl | |- ( A. x A. y ph -> A. y A. x ph ) |