 Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ax-c11n Structured version   Visualization version   GIF version

Axiom ax-c11n 32987
 Description: Axiom of Quantifier Substitution. One of the equality and substitution axioms of predicate calculus with equality. Appears as Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). The original version of this axiom was ax-c11 32986 and was replaced with this shorter ax-c11n 32987 ("n" for "new") in May 2008. The old axiom is proved from this one as theorem axc11 2301. Conversely, this axiom is proved from ax-c11 32986 as theorem axc11nfromc11 33025. This axiom was proved redundant in July 2015. See theorem axc11n 2294. This axiom is obsolete and should no longer be used. It is proved above as theorem axc11n 2294. (Contributed by NM, 16-May-2008.) (New usage is discouraged.)
Assertion
Ref Expression
ax-c11n (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Detailed syntax breakdown of Axiom ax-c11n
StepHypRef Expression
1 vx . . . 4 setvar 𝑥
2 vy . . . 4 setvar 𝑦
31, 2weq 1860 . . 3 wff 𝑥 = 𝑦
43, 1wal 1472 . 2 wff 𝑥 𝑥 = 𝑦
52, 1weq 1860 . . 3 wff 𝑦 = 𝑥
65, 2wal 1472 . 2 wff 𝑦 𝑦 = 𝑥
74, 6wi 4 1 wff (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
 Colors of variables: wff setvar class This axiom is referenced by:  axc11-o  33050
 Copyright terms: Public domain W3C validator