Theorem mo2 2233

Description: Alternate definition of "at most one." (Contributed by NM, 8-Mar-1995.)

Hypothesis

Ref	Expression
mo2.1	⊢ Ⅎyφ

Assertion

Ref	Expression
mo2	⊢ (∃*xφ ↔ ∃y∀x(φ → x = y))

Distinct variable group:   x,y

Allowed substitution hints:   φ(x,y)

Proof of Theorem mo2

Step	Hyp	Ref	Expression
1		df-mo 2209	. 2 ⊢ (∃*xφ ↔ (∃xφ → ∃!xφ))
2		alnex 1543	. . . . 5 ⊢ (∀x ¬ φ ↔ ¬ ∃xφ)
3		pm2.21 100	. . . . . . 7 ⊢ (¬ φ → (φ → x = y))
4	3	alimi 1559	. . . . . 6 ⊢ (∀x ¬ φ → ∀x(φ → x = y))
5		19.8a 1756	. . . . . 6 ⊢ (∀x(φ → x = y) → ∃y∀x(φ → x = y))
6	4, 5	syl 15	. . . . 5 ⊢ (∀x ¬ φ → ∃y∀x(φ → x = y))
7	2, 6	sylbir 204	. . . 4 ⊢ (¬ ∃xφ → ∃y∀x(φ → x = y))
8		mo2.1	. . . . 5 ⊢ Ⅎyφ
9	8	eumo0 2228	. . . 4 ⊢ (∃!xφ → ∃y∀x(φ → x = y))
10	7, 9	ja 153	. . 3 ⊢ ((∃xφ → ∃!xφ) → ∃y∀x(φ → x = y))
11	8	eu3 2230	. . . 4 ⊢ (∃!xφ ↔ (∃xφ ∧ ∃y∀x(φ → x = y)))
12	11	simplbi2com 1374	. . 3 ⊢ (∃y∀x(φ → x = y) → (∃xφ → ∃!xφ))
13	10, 12	impbii 180	. 2 ⊢ ((∃xφ → ∃!xφ) ↔ ∃y∀x(φ → x = y))
14	1, 13	bitri 240	1 ⊢ (∃*xφ ↔ ∃y∀x(φ → x = y))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544  ∃!weu 2204  ∃*wmo 2205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209

This theorem is referenced by:  sbmo  2234  mo3  2235  eu5  2242  moim  2250  morimv  2252  moanim  2260  mo2icl  3016  rmo2  3132  dffun3  5121  dffun6f  5124