Theorem exmodc 2069

Description: If existence is decidable, something exists or at most one exists. (Contributed by Jim Kingdon, 30-Jun-2018.)

Assertion

Ref	Expression
exmodc	⊢ (DECID ∃𝑥𝜑 → (∃𝑥𝜑 ∨ ∃*𝑥𝜑))

Proof of Theorem exmodc

Step	Hyp	Ref	Expression
1		df-dc 830	. 2 ⊢ (DECID ∃𝑥𝜑 ↔ (∃𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
2		pm2.21 612	. . . 4 ⊢ (¬ ∃𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑))
3		df-mo 2023	. . . 4 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
4	2, 3	sylibr 133	. . 3 ⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)
5	4	orim2i 756	. 2 ⊢ ((∃𝑥𝜑 ∨ ¬ ∃𝑥𝜑) → (∃𝑥𝜑 ∨ ∃*𝑥𝜑))
6	1, 5	sylbi 120	1 ⊢ (DECID ∃𝑥𝜑 → (∃𝑥𝜑 ∨ ∃*𝑥𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 703  DECID wdc 829  ∃wex 1485  ∃!weu 2019  ∃*wmo 2020

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 610  ax-io 704

This theorem depends on definitions:  df-bi 116  df-dc 830  df-mo 2023

This theorem is referenced by: (None)