Description: Weak version of ax-12 from which we can prove any ax-12 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. An instance of the first hypothesis will normally require that x and y be distinct (unless x does not occur in ph ). For an example of how the hypotheses can be eliminated when we substitute an expression without wff variables for ph , see ax12wdemo . (Contributed by NM, 10-Apr-2017)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | ax12w.1 | |- ( x = y -> ( ph <-> ps ) ) |
|
| ax12w.2 | |- ( y = z -> ( ph <-> ch ) ) |
||
| Assertion | ax12w | |- ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ax12w.1 | |- ( x = y -> ( ph <-> ps ) ) |
|
| 2 | ax12w.2 | |- ( y = z -> ( ph <-> ch ) ) |
|
| 3 | 2 | spw | |- ( A. y ph -> ph ) |
| 4 | 1 | ax12wlem | |- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) |
| 5 | 3 4 | syl5 | |- ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) ) |