Theorem bnj1304 30144

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1304.1	⊢ (𝜑 → ∃𝑥𝜓)
bnj1304.2	⊢ (𝜓 → 𝜒)
bnj1304.3	⊢ (𝜓 → ¬ 𝜒)

Assertion

Ref	Expression
bnj1304	⊢ ¬ 𝜑

Proof of Theorem bnj1304

Step	Hyp	Ref	Expression
1		notnotb 303	. . . 4 ⊢ (∀𝑥(𝜒 ∨ ¬ 𝜒) ↔ ¬ ¬ ∀𝑥(𝜒 ∨ ¬ 𝜒))
2		notnotb 303	. . . . . . . 8 ⊢ (𝜒 ↔ ¬ ¬ 𝜒)
3	2	anbi2i 726	. . . . . . 7 ⊢ ((¬ 𝜒 ∧ 𝜒) ↔ (¬ 𝜒 ∧ ¬ ¬ 𝜒))
4	3	exbii 1764	. . . . . 6 ⊢ (∃𝑥(¬ 𝜒 ∧ 𝜒) ↔ ∃𝑥(¬ 𝜒 ∧ ¬ ¬ 𝜒))
5		ioran 510	. . . . . . 7 ⊢ (¬ (𝜒 ∨ ¬ 𝜒) ↔ (¬ 𝜒 ∧ ¬ ¬ 𝜒))
6	5	exbii 1764	. . . . . 6 ⊢ (∃𝑥 ¬ (𝜒 ∨ ¬ 𝜒) ↔ ∃𝑥(¬ 𝜒 ∧ ¬ ¬ 𝜒))
7		exnal 1744	. . . . . 6 ⊢ (∃𝑥 ¬ (𝜒 ∨ ¬ 𝜒) ↔ ¬ ∀𝑥(𝜒 ∨ ¬ 𝜒))
8	4, 6, 7	3bitr2ri 288	. . . . 5 ⊢ (¬ ∀𝑥(𝜒 ∨ ¬ 𝜒) ↔ ∃𝑥(¬ 𝜒 ∧ 𝜒))
9	8	notbii 309	. . . 4 ⊢ (¬ ¬ ∀𝑥(𝜒 ∨ ¬ 𝜒) ↔ ¬ ∃𝑥(¬ 𝜒 ∧ 𝜒))
10		exancom 1774	. . . . 5 ⊢ (∃𝑥(¬ 𝜒 ∧ 𝜒) ↔ ∃𝑥(𝜒 ∧ ¬ 𝜒))
11	10	notbii 309	. . . 4 ⊢ (¬ ∃𝑥(¬ 𝜒 ∧ 𝜒) ↔ ¬ ∃𝑥(𝜒 ∧ ¬ 𝜒))
12	1, 9, 11	3bitri 285	. . 3 ⊢ (∀𝑥(𝜒 ∨ ¬ 𝜒) ↔ ¬ ∃𝑥(𝜒 ∧ ¬ 𝜒))
13		exmid 430	. . 3 ⊢ (𝜒 ∨ ¬ 𝜒)
14	12, 13	mpgbi 1716	. 2 ⊢ ¬ ∃𝑥(𝜒 ∧ ¬ 𝜒)
15		bnj1304.1	. . 3 ⊢ (𝜑 → ∃𝑥𝜓)
16		bnj1304.2	. . . 4 ⊢ (𝜓 → 𝜒)
17		bnj1304.3	. . . 4 ⊢ (𝜓 → ¬ 𝜒)
18	16, 17	jca 553	. . 3 ⊢ (𝜓 → (𝜒 ∧ ¬ 𝜒))
19	15, 18	bnj593 30069	. 2 ⊢ (𝜑 → ∃𝑥(𝜒 ∧ ¬ 𝜒))
20	14, 19	mto 187	1 ⊢ ¬ 𝜑

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383  ∀wal 1473  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ex 1696

This theorem is referenced by:  bnj1204  30334  bnj1279  30340  bnj1311  30346  bnj1312  30380