Theorem exists2 1457

Description: A condition implying that at least two things exist.

Assertion

Ref	Expression
exists2	⊢ ((∃xφ ⋀ ∃x ¬ φ) → ¬ ∃!x x = x)

Proof of Theorem exists2

Step	Hyp	Ref	Expression
1		exists1 1456	. . 3 ⊢ (∃!x x = x ↔ ∀x x = y)
2		pm3.24 657	. . . 4 ⊢ ¬ (φ ⋀ ¬ φ)
3		ax-16 1209	. . . . . . 7 ⊢ (∀x x = y → (φ → ∀xφ))
4	3	a5i 988	. . . . . 6 ⊢ (∀x x = y → ∀x(φ → ∀xφ))
5		19.9t 1034	. . . . . 6 ⊢ (∀x(φ → ∀xφ) → (∃xφ → φ))
6	4, 5	syl 10	. . . . 5 ⊢ (∀x x = y → (∃xφ → φ))
7		ax-16 1209	. . . . . . 7 ⊢ (∀x x = y → (¬ φ → ∀x ¬ φ))
8	7	a5i 988	. . . . . 6 ⊢ (∀x x = y → ∀x(¬ φ → ∀x ¬ φ))
9		19.9t 1034	. . . . . 6 ⊢ (∀x(¬ φ → ∀x ¬ φ) → (∃x ¬ φ → ¬ φ))
10	8, 9	syl 10	. . . . 5 ⊢ (∀x x = y → (∃x ¬ φ → ¬ φ))
11	6, 10	anim12d 557	. . . 4 ⊢ (∀x x = y → ((∃xφ ⋀ ∃x ¬ φ) → (φ ⋀ ¬ φ)))
12	2, 11	mtoi 107	. . 3 ⊢ (∀x x = y → ¬ (∃xφ ⋀ ∃x ¬ φ))
13	1, 12	sylbi 199	. 2 ⊢ (∃!x x = x → ¬ (∃xφ ⋀ ∃x ¬ φ))
14	13	con2i 97	1 ⊢ ((∃xφ ⋀ ∃x ¬ φ) → ¬ ∃!x x = x)

Syntax hints:  ¬ wn 2   → wi 3   ⋀ wa 223  ∀wal 953   = wceq 955  ∃wex 979  ∃!weu 1379

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-10 965  ax-12 967  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-eu 1381

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