Theorem exists2OLD 2694

Description: Obsolete version of exists2 2693 as of 4-Mar-2023. (Contributed by NM, 10-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
exists2OLD	⊢ ((∃𝑥𝜑 ∧ ∃𝑥 ¬ 𝜑) → ¬ ∃!𝑥 𝑥 = 𝑥)

Proof of Theorem exists2OLD

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfeu1 2609	. . . . . 6 ⊢ Ⅎ𝑥∃!𝑥 𝑥 = 𝑥
2		nfa1 2145	. . . . . 6 ⊢ Ⅎ𝑥∀𝑥𝜑
3		exists1 2692	. . . . . . 7 ⊢ (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)
4		axc16 2234	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
5	3, 4	sylbi 209	. . . . . 6 ⊢ (∃!𝑥 𝑥 = 𝑥 → (𝜑 → ∀𝑥𝜑))
6	1, 2, 5	exlimd 2204	. . . . 5 ⊢ (∃!𝑥 𝑥 = 𝑥 → (∃𝑥𝜑 → ∀𝑥𝜑))
7	6	com12 32	. . . 4 ⊢ (∃𝑥𝜑 → (∃!𝑥 𝑥 = 𝑥 → ∀𝑥𝜑))
8		alex 1869	. . . 4 ⊢ (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
9	7, 8	syl6ib 243	. . 3 ⊢ (∃𝑥𝜑 → (∃!𝑥 𝑥 = 𝑥 → ¬ ∃𝑥 ¬ 𝜑))
10	9	con2d 132	. 2 ⊢ (∃𝑥𝜑 → (∃𝑥 ¬ 𝜑 → ¬ ∃!𝑥 𝑥 = 𝑥))
11	10	imp 397	1 ⊢ ((∃𝑥𝜑 ∧ ∃𝑥 ¬ 𝜑) → ¬ ∃!𝑥 𝑥 = 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 386  ∀wal 1599  ∃wex 1823  ∃!weu 2586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-10 2135  ax-11 2150  ax-12 2163

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-mo 2551  df-eu 2587

This theorem is referenced by: (None)