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

Theorem ax5ALT 33639
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 1841 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 1719, ax-c4 33616, ax-c5 33615, ax-11 2036, ax-c7 33617, ax-7 1937, ax-c9 33622, ax-c10 33618, ax-c11 33619, ax-8 1994, ax-9 2001, ax-c14 33623, ax-c15 33621, and ax-c16 33624: in that system, we can derive any instance of ax-5 1841 not containing wff variables by induction on formula length, using ax5eq 33664 and ax5el 33669 for the basis together with hbn 2147, hbal 2038, and hbim 2128. 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.)

Ref Expression
ax5ALT (𝜑 → ∀𝑥𝜑)
Distinct variable group:   𝜑,𝑥

Proof of Theorem ax5ALT
StepHypRef Expression
1 ax-5 1841 1 (𝜑 → ∀𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1478
This theorem was proved from axioms:  ax-5 1841
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator