MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax-12 Structured version   Visualization version   GIF version

Axiom ax-12 2167
Description: Axiom of Substitution. One of the 5 equality axioms of predicate calculus. The final consequent 𝑥(𝑥 = 𝑦𝜑) is a way of expressing "𝑦 substituted for 𝑥 in wff 𝜑 " (cf. sb6 2084). 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.

The original version of this axiom was ax-c15 35907 and was replaced with this shorter ax-12 2167 in Jan. 2007. The old axiom is proved from this one as theorem axc15 2438. Conversely, this axiom is proved from ax-c15 35907 as theorem ax12 2440.

Juha Arpiainen proved the metalogical independence of this axiom (in the form of the older axiom ax-c15 35907) from the others on 19-Jan-2006. See item 9a at https://us.metamath.org/award2003.html 35907.

See ax12v 2168 and ax12v2 2169 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions.

This axiom scheme is logically redundant (see ax12w 2128) but is used as an auxiliary axiom scheme to achieve scheme completeness. (Contributed by NM, 22-Jan-2007.) (New usage is discouraged.)

Assertion
Ref Expression
ax-12 (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Detailed syntax breakdown of Axiom ax-12
StepHypRef Expression
1 vx . . 3 setvar 𝑥
2 vy . . 3 setvar 𝑦
31, 2weq 1955 . 2 wff 𝑥 = 𝑦
4 wph . . . 4 wff 𝜑
54, 2wal 1526 . . 3 wff 𝑦𝜑
63, 4wi 4 . . . 4 wff (𝑥 = 𝑦𝜑)
76, 1wal 1526 . . 3 wff 𝑥(𝑥 = 𝑦𝜑)
85, 7wi 4 . 2 wff (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
93, 8wi 4 1 wff (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
Colors of variables: wff setvar class
This axiom is referenced by:  ax12v  2168  equs5aALT  2377  equs5eALT  2378  axc11r  2379
  Copyright terms: Public domain W3C validator