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

Theorem ax5ALT 32710
 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 1789 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 1698, ax-c4 32689, ax-c5 32688, ax-11 1970, ax-c7 32690, ax-7 1883, ax-c9 32695, ax-c10 32691, ax-c11 32692, ax-8 1939, ax-9 1946, ax-c14 32696, ax-c15 32694, and ax-c16 32697: in that system, we can derive any instance of ax-5 1789 not containing wff variables by induction on formula length, using ax5eq 32736 and ax5el 32741 for the basis together hbn 2030, hbal 1972, and hbim 2058. 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 1789 1 (𝜑 → ∀𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1466 This theorem was proved from axioms:  ax-5 1789 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator