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Axiom ax-17 1634
 Description: Axiom to quantify a variable over a formula in which it does not occur. Axiom C5 in [Megill] p. 444 (p. 11 of the preprint). Also appears as Axiom B6 (p. 75) of system S2 of [Tarski] p. 77 and Axiom C5-1 of [Monk2] p. 113. This axiom is logically redundant in the (logically complete) predicate calculus axiom system consisting of ax-gen 1622, ax-4 1637 through ax-9 1624, ax-10o 1810, and ax-12 1627 through ax-16 1883: in that system, we can derive any instance of ax-17 1634 not containing wff variables by induction on formula length, using ax17eq 1884 and ax17el 2045 for the basis together hbn 1669, hbal 1670, and hbim 1672. However, if we omit this axiom, our development would be quite inconvenient since we could work only with specific instances of wffs containing no wff variables - this axiom introduces the concept of a set variable not occurring in a wff (as opposed to just two set variables being distinct).
Assertion
Ref Expression
ax-17
Distinct variable group:   ,

Detailed syntax breakdown of Axiom ax-17
StepHypRef Expression
1 wph . 2
2 vx . . 3
31, 2wal 1613 . 2
41, 3wi 3 1