MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax5ALT Unicode version

Theorem ax5ALT 2236
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 1704 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 1618, ax-c4 2215, ax-c5 2214, ax-11 1842, ax-c7 2216, ax-7 1790, ax-c9 2221, ax-c10 2217, ax-c11 2218, ax-8 1820, ax-9 1822, ax-c14 2222, ax-c15 2220, and ax-c16 2223: in that system, we can derive any instance of ax-5 1704 not containing wff variables by induction on formula length, using ax5eq 2262 and ax5el 2267 for the basis together hbn 1895, hbal 1844, and hbim 1922. 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 1704 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  A.wal 1393
This theorem was proved from axioms:  ax-5 1704
  Copyright terms: Public domain W3C validator