Theorem alexim 1638

Description: One direction of theorem 19.6 of [Margaris] p. 89. The converse holds given a decidability condition, as seen at alexdc 1612. (Contributed by Jim Kingdon, 2-Jul-2018.)

Assertion

Ref	Expression
alexim	⊢ (∀𝑥𝜑 → ¬ ∃𝑥 ¬ 𝜑)

Proof of Theorem alexim

Step	Hyp	Ref	Expression
1		pm2.24 616	. . . . 5 ⊢ (𝜑 → (¬ 𝜑 → ⊥))
2	1	alimi 1448	. . . 4 ⊢ (∀𝑥𝜑 → ∀𝑥(¬ 𝜑 → ⊥))
3		exim 1592	. . . 4 ⊢ (∀𝑥(¬ 𝜑 → ⊥) → (∃𝑥 ¬ 𝜑 → ∃𝑥⊥))
4	2, 3	syl 14	. . 3 ⊢ (∀𝑥𝜑 → (∃𝑥 ¬ 𝜑 → ∃𝑥⊥))
5		nfv 1521	. . . 4 ⊢ Ⅎ𝑥⊥
6	5	19.9 1637	. . 3 ⊢ (∃𝑥⊥ ↔ ⊥)
7	4, 6	syl6ib 160	. 2 ⊢ (∀𝑥𝜑 → (∃𝑥 ¬ 𝜑 → ⊥))
8		dfnot 1366	. 2 ⊢ (¬ ∃𝑥 ¬ 𝜑 ↔ (∃𝑥 ¬ 𝜑 → ⊥))
9	7, 8	sylibr 133	1 ⊢ (∀𝑥𝜑 → ¬ ∃𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1346  ⊥wfal 1353  ∃wex 1485

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-in1 609  ax-in2 610  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454

This theorem is referenced by:  exnalim  1639  exists2  2116