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Axiom ax-12 2045
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 2427). 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 33993 and was replaced with this shorter ax-12 2045 in Jan. 2007. The old axiom is proved from this one as theorem axc15 2301. Conversely, this axiom is proved from ax-c15 33993 as theorem ax12 2302.

Juha Arpiainen proved the metalogical independence of this axiom (in the form of the older axiom ax-c15 33993) from the others on 19-Jan-2006. See item 9a at http://us.metamath.org/award2003.html.

See ax12v 2046 and ax12v2 2047 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions.

This axiom scheme is logically redundant (see ax12w 2008) 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 1872 . 2 wff 𝑥 = 𝑦
4 wph . . . 4 wff 𝜑
54, 2wal 1479 . . 3 wff 𝑦𝜑
63, 4wi 4 . . . 4 wff (𝑥 = 𝑦𝜑)
76, 1wal 1479 . . 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  2046  ax12vOLD  2048  ax12vOLDOLD  2049  equs5aALT  2175  equs5eALT  2176  axc11r  2185  axc15OLD  2342
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