Theorem qexmid 2194

Description: Quantified excluded middle (see exmid 894). Also known as the drinker paradox (if 𝜑(𝑥) is interpreted as "𝑥 drinks", then this theorem tells that there exists a person such that, if this person drinks, then everyone drinks). Exercise 9.2a of Boolos, p. 111, Computability and Logic. (Contributed by NM, 10-Dec-2000.)

Assertion

Ref	Expression
qexmid	⊢ ∃𝑥(𝜑 → ∀𝑥𝜑)

Proof of Theorem qexmid

Step	Hyp	Ref	Expression
1		19.8a 2182	. 2 ⊢ (∀𝑥𝜑 → ∃𝑥∀𝑥𝜑)
2	1	19.35ri 1879	1 ⊢ ∃𝑥(𝜑 → ∀𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1538  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-ex 1780

This theorem is referenced by: (None)