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

Theorem ax5ALT 33210
 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 1827 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 1713, ax-c4 33187, ax-c5 33186, ax-11 2021, ax-c7 33188, ax-7 1922, ax-c9 33193, ax-c10 33189, ax-c11 33190, ax-8 1979, ax-9 1986, ax-c14 33194, ax-c15 33192, and ax-c16 33195: in that system, we can derive any instance of ax-5 1827 not containing wff variables by induction on formula length, using ax5eq 33235 and ax5el 33240 for the basis together with hbn 2131, hbal 2023, and hbim 2112. 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 (𝜑 → ∀𝑥𝜑)
Distinct variable group:   𝜑,𝑥

Proof of Theorem ax5ALT
StepHypRef Expression
1 ax-5 1827 1 (𝜑 → ∀𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1473 This theorem was proved from axioms:  ax-5 1827 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator