Description: Inference form of Theorem
19.23 of [Margaris] p. 90, see 19.231910.
See exlimi1912 for a more general version requiring more
axioms.
This inference, along with its many variants such as rexlimdv2947, 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. E.x(x) where (x) 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
exlimiv1722 to arrive at . Finally, we separately
prove and detach it
with modus ponens ax-mp5 to arrive at
the final theorem . (Contributed by NM,
21-Jun-1993.) Remove
dependencies on ax-61747 and ax-81820. (Revised by Wolf Lammen,
4-Dec-2017.)