**Description:** Axiom ax-c15 was the original version of ax-12 , before it was
discovered (in Jan. 2007) that the shorter ax-12 could replace it. It
appears as Axiom scheme C15' in Megill p. 448 (p. 16 of the preprint).
It is based on Lemma 16 of Tarski p. 70 and Axiom C8 of Monk2 p. 105,
from which it can be proved by cases. To understand this theorem more
easily, think of " -. A. x x = y -> ..." as informally meaning "if
x and y are distinct variables then..." The antecedent becomes
false if the same variable is substituted for x and y , ensuring
the theorem is sound whenever this is the case. In some later theorems,
we call an antecedent of the form -. A. x x = y a "distinctor."

Interestingly, if the wff expression substituted for ph contains no
wff variables, the resulting statement*can* be proved without invoking
this axiom. This means that even though this axiom is*metalogically*
independent from the others, it is not*logically* independent.
Specifically, we can prove any wff-variable-free instance of axiom
ax-c15 (from which the ax-12 instance follows by theorem ax12 .)
The proof is by induction on formula length, using ax12eq and ax12el for the basis steps and ax12indn , ax12indi , and ax12inda for the
induction steps. (This paragraph is true provided we use ax-c11 in
place of ax-c11n .)

This axiom is obsolete and should no longer be used. It is proved above as theorem axc15 , which should be used instead. (Contributed by NM, 14-May-1993) (New usage is discouraged.)

Ref | Expression | ||
---|---|---|---|

Assertion | ax-c15 | $${\u22a2}\neg \forall {x}\phantom{\rule{.4em}{0ex}}{x}={y}\to \left({x}={y}\to \left({\phi}\to \forall {x}\phantom{\rule{.4em}{0ex}}\left({x}={y}\to {\phi}\right)\right)\right)$$ |

Step | Hyp | Ref | Expression |
---|---|---|---|

0 | vx | $${setvar}{x}$$ | |

1 | 0 | cv | $${setvar}{x}$$ |

2 | vy | $${setvar}{y}$$ | |

3 | 2 | cv | $${setvar}{y}$$ |

4 | 1 3 | wceq | $${wff}{x}={y}$$ |

5 | 4 0 | wal | $${wff}\forall {x}\phantom{\rule{.4em}{0ex}}{x}={y}$$ |

6 | 5 | wn | $${wff}\neg \forall {x}\phantom{\rule{.4em}{0ex}}{x}={y}$$ |

7 | wph | $${wff}{\phi}$$ | |

8 | 4 7 | wi | $${wff}\left({x}={y}\to {\phi}\right)$$ |

9 | 8 0 | wal | $${wff}\forall {x}\phantom{\rule{.4em}{0ex}}\left({x}={y}\to {\phi}\right)$$ |

10 | 7 9 | wi | $${wff}\left({\phi}\to \forall {x}\phantom{\rule{.4em}{0ex}}\left({x}={y}\to {\phi}\right)\right)$$ |

11 | 4 10 | wi | $${wff}\left({x}={y}\to \left({\phi}\to \forall {x}\phantom{\rule{.4em}{0ex}}\left({x}={y}\to {\phi}\right)\right)\right)$$ |

12 | 6 11 | wi | $${wff}\left(\neg \forall {x}\phantom{\rule{.4em}{0ex}}{x}={y}\to \left({x}={y}\to \left({\phi}\to \forall {x}\phantom{\rule{.4em}{0ex}}\left({x}={y}\to {\phi}\right)\right)\right)\right)$$ |