Description: Inference form of Theorem
19.23 of [Margaris] p. 90, see 19.23 2004.
See exlimi 2006 for a more general version requiring more
axioms.
This inference, along with its many variants such as rexlimdv 2889, 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 1787 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, 21-Jun-1993.) Remove
dependencies on ax-6 1816 and ax-8 1900. (Revised by Wolf Lammen,
4-Dec-2017.) |