Theorem alexim 1536

Description: One direction of theorem 19.6 of [Margaris] p. 89. The converse holds given a decidability condition, as seen at alexdc 1510. (Contributed by Jim Kingdon, 2-Jul-2018.)

Assertion

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

Proof of Theorem alexim

Step	Hyp	Ref	Expression
1		pm2.24 551	. . . . 5 ⊢ (𝜑 → (¬ 𝜑 → ⊥))
2	1	alimi 1344	. . . 4 ⊢ (∀𝑥𝜑 → ∀𝑥(¬ 𝜑 → ⊥))
3		exim 1490	. . . 4 ⊢ (∀𝑥(¬ 𝜑 → ⊥) → (∃𝑥 ¬ 𝜑 → ∃𝑥⊥))
4	2, 3	syl 14	. . 3 ⊢ (∀𝑥𝜑 → (∃𝑥 ¬ 𝜑 → ∃𝑥⊥))
5		nfv 1421	. . . 4 ⊢ Ⅎ𝑥⊥
6	5	19.9 1535	. . 3 ⊢ (∃𝑥⊥ ↔ ⊥)
7	4, 6	syl6ib 150	. 2 ⊢ (∀𝑥𝜑 → (∃𝑥 ¬ 𝜑 → ⊥))
8		dfnot 1262	. 2 ⊢ (¬ ∃𝑥 ¬ 𝜑 ↔ (∃𝑥 ¬ 𝜑 → ⊥))
9	7, 8	sylibr 137	1 ⊢ (∀𝑥𝜑 → ¬ ∃𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1241  ⊥wfal 1248  ∃wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350

This theorem is referenced by:  exnalim  1537  exists2  1997