| Description: Inference from Theorem
19.23 of [Margaris] p. 90, see 19.23 1841.
This inference, along with our many variants such as rexlimdv 2830, is
used to implement a metatheorem called "Rule C" that is given
in many
logic textbooks. See, for example, Rule C in [Mendelson] p. 81, Rule C
in [Margaris] p. 40, or Rule C in
Hirst and Hirst's A Primer for Logic
and Proof p. 59 (PDF p. 65) at
http://www.appstate.edu/~hirstjl/primer/hirst.pdf.
In informal
proofs, the statement "Let  be an element such that..." almost
always means an implicit application of Rule C.
In essence, Rule C states that if we can prove that some element 
exists satisfying a wff, i.e.     where has
free, then we
can use as a hypothesis for the proof
where is a new
(fictitious) constant not appearing previously in
the proof, nor in any axioms used, nor in the theorem to be proved. The
purpose of Rule C is to get rid of the existential quantifier.
We cannot do this in Metamath directly. Instead, we use the original
(containing ) as an
antecedent for the main part of the
proof. We eventually arrive at 
 where is the
theorem to be proved and does not contain . Then we apply
exlimiv 1687 to arrive at . Finally, we separately
prove and detach it with modus ponens ax-mp 5
to arrive at
the final theorem . (Contributed by NM, 5-Aug-1993.) (Revised
by Wolf Lammen to remove dependency on ax-6 and ax-8,
4-Dec-2017.) |
