Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ax5ALT Structured version   Unicode version

Theorem ax5ALT 32402
 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 theorem simply repeats ax-5 1749 so that we can include the following note, which applies only to the obsolete axiomatization.) This axiom is logically redundant in the (logically complete) predicate calculus axiom system consisting of ax-gen 1666, ax-c4 32381, ax-c5 32380, ax-11 1893, ax-c7 32382, ax-7 1840, ax-c9 32387, ax-c10 32383, ax-c11 32384, ax-8 1871, ax-9 1873, ax-c14 32388, ax-c15 32386, and ax-c16 32389: in that system, we can derive any instance of ax-5 1749 not containing wff variables by induction on formula length, using ax5eq 32428 and ax5el 32433 for the basis together hbn 1951, hbal 1895, and hbim 1979. 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 setvar variable not occurring in a wff (as opposed to just two setvar variables being distinct). (Contributed by NM, 19-Aug-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
ax5ALT
Distinct variable group:   ,

Proof of Theorem ax5ALT
StepHypRef Expression
1 ax-5 1749 1
 Colors of variables: wff setvar class Syntax hints:   wi 4  wal 1436 This theorem was proved from axioms:  ax-5 1749 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator