Theorem bnj1304

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

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

Proof

Step	Hyp	Ref	Expression
1		bnj1304.1	⊢ ( 𝜑 → ∃ 𝑥 𝜓 )
2		bnj1304.2	⊢ ( 𝜓 → 𝜒 )
3		bnj1304.3	⊢ ( 𝜓 → ¬ 𝜒 )
4		notnotb	⊢ ( ∀ 𝑥 ( 𝜒 ∨ ¬ 𝜒 ) ↔ ¬ ¬ ∀ 𝑥 ( 𝜒 ∨ ¬ 𝜒 ) )
5		notnotb	⊢ ( 𝜒 ↔ ¬ ¬ 𝜒 )
6	5	anbi2i	⊢ ( ( ¬ 𝜒 ∧ 𝜒 ) ↔ ( ¬ 𝜒 ∧ ¬ ¬ 𝜒 ) )
7	6	exbii	⊢ ( ∃ 𝑥 ( ¬ 𝜒 ∧ 𝜒 ) ↔ ∃ 𝑥 ( ¬ 𝜒 ∧ ¬ ¬ 𝜒 ) )
8		ioran	⊢ ( ¬ ( 𝜒 ∨ ¬ 𝜒 ) ↔ ( ¬ 𝜒 ∧ ¬ ¬ 𝜒 ) )
9	8	exbii	⊢ ( ∃ 𝑥 ¬ ( 𝜒 ∨ ¬ 𝜒 ) ↔ ∃ 𝑥 ( ¬ 𝜒 ∧ ¬ ¬ 𝜒 ) )
10		exnal	⊢ ( ∃ 𝑥 ¬ ( 𝜒 ∨ ¬ 𝜒 ) ↔ ¬ ∀ 𝑥 ( 𝜒 ∨ ¬ 𝜒 ) )
11	7 9 10	3bitr2ri	⊢ ( ¬ ∀ 𝑥 ( 𝜒 ∨ ¬ 𝜒 ) ↔ ∃ 𝑥 ( ¬ 𝜒 ∧ 𝜒 ) )
12	11	notbii	⊢ ( ¬ ¬ ∀ 𝑥 ( 𝜒 ∨ ¬ 𝜒 ) ↔ ¬ ∃ 𝑥 ( ¬ 𝜒 ∧ 𝜒 ) )
13		exancom	⊢ ( ∃ 𝑥 ( ¬ 𝜒 ∧ 𝜒 ) ↔ ∃ 𝑥 ( 𝜒 ∧ ¬ 𝜒 ) )
14	13	notbii	⊢ ( ¬ ∃ 𝑥 ( ¬ 𝜒 ∧ 𝜒 ) ↔ ¬ ∃ 𝑥 ( 𝜒 ∧ ¬ 𝜒 ) )
15	4 12 14	3bitri	⊢ ( ∀ 𝑥 ( 𝜒 ∨ ¬ 𝜒 ) ↔ ¬ ∃ 𝑥 ( 𝜒 ∧ ¬ 𝜒 ) )
16		exmid	⊢ ( 𝜒 ∨ ¬ 𝜒 )
17	15 16	mpgbi	⊢ ¬ ∃ 𝑥 ( 𝜒 ∧ ¬ 𝜒 )
18	2 3	jca	⊢ ( 𝜓 → ( 𝜒 ∧ ¬ 𝜒 ) )
19	1 18	bnj593	⊢ ( 𝜑 → ∃ 𝑥 ( 𝜒 ∧ ¬ 𝜒 ) )
20	17 19	mto	⊢ ¬ 𝜑