Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ax-c9 Structured version   Visualization version   GIF version

Axiom ax-c9 36158
Description: Axiom of Quantifier Introduction. One of the equality and substitution axioms of predicate calculus with equality. Informally, it says that whenever 𝑧 is distinct from 𝑥 and 𝑦, and 𝑥 = 𝑦 is true, then 𝑥 = 𝑦 quantified with 𝑧 is also true. In other words, 𝑧 is irrelevant to the truth of 𝑥 = 𝑦. Axiom scheme C9' in [Megill] p. 448 (p. 16 of the preprint). It apparently does not otherwise appear in the literature but is easily proved from textbook predicate calculus by cases.

This axiom is obsolete and should no longer be used. It is proved above as theorem axc9 2402. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)

Assertion
Ref Expression
ax-c9 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))

Detailed syntax breakdown of Axiom ax-c9
StepHypRef Expression
1 vz . . . . 5 setvar 𝑧
2 vx . . . . 5 setvar 𝑥
31, 2weq 1965 . . . 4 wff 𝑧 = 𝑥
43, 1wal 1536 . . 3 wff 𝑧 𝑧 = 𝑥
54wn 3 . 2 wff ¬ ∀𝑧 𝑧 = 𝑥
6 vy . . . . . 6 setvar 𝑦
71, 6weq 1965 . . . . 5 wff 𝑧 = 𝑦
87, 1wal 1536 . . . 4 wff 𝑧 𝑧 = 𝑦
98wn 3 . . 3 wff ¬ ∀𝑧 𝑧 = 𝑦
102, 6weq 1965 . . . 4 wff 𝑥 = 𝑦
1110, 1wal 1536 . . . 4 wff 𝑧 𝑥 = 𝑦
1210, 11wi 4 . . 3 wff (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
139, 12wi 4 . 2 wff (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
145, 13wi 4 1 wff (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
Colors of variables: wff setvar class
This axiom is referenced by:  equid1  36167  hbae-o  36171  ax13fromc9  36174  hbequid  36177  equid1ALT  36193  dvelimf-o  36197  ax5eq  36200
  Copyright terms: Public domain W3C validator