Description: Inference that has ax-12 (without A. y ) as its conclusion. Uses only Tarski's FOL axiom schemes. The hypotheses may be eliminable without using ax-12 in special cases. Proof similar to Lemma 16 of Tarski p. 70. (Contributed by NM, 20-May-2008)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | ax12i.1 | |- ( x = y -> ( ph <-> ps ) ) |
|
| ax12i.2 | |- ( ps -> A. x ps ) |
||
| Assertion | ax12i | |- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ax12i.1 | |- ( x = y -> ( ph <-> ps ) ) |
|
| 2 | ax12i.2 | |- ( ps -> A. x ps ) |
|
| 3 | 1 | biimprcd | |- ( ps -> ( x = y -> ph ) ) |
| 4 | 2 3 | alrimih | |- ( ps -> A. x ( x = y -> ph ) ) |
| 5 | 1 4 | syl6bi | |- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) |