Theorem mo2 1026

Description: Alternate definition of "at most one".

Hypothesis

Ref	Expression
mo2.1	⊢ (φ → ∀yφ)

Assertion

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

Distinct variable group(s):   x,y

Proof of Theorem mo2

Step	Hyp	Ref	Expression
1		df-mo 1010	. 2 ⊢ (∃*xφ ↔ (∃xφ → ∃!xφ))
2		alnex 716	. . . . 5 ⊢ (∀x ¬ φ ↔ ¬ ∃xφ)
3		pm2.21 71	. . . . . . 7 ⊢ (¬ φ → (φ → x = y))
4	3	19.20i 691	. . . . . 6 ⊢ (∀x ¬ φ → ∀x(φ → x = y))
5		19.8a 712	. . . . . 6 ⊢ (∀x(φ → x = y) → ∃y∀x(φ → x = y))
6	4, 5	syl 12	. . . . 5 ⊢ (∀x ¬ φ → ∃y∀x(φ → x = y))
7	2, 6	sylbir 176	. . . 4 ⊢ (¬ ∃xφ → ∃y∀x(φ → x = y))
8		mo2.1	. . . . 5 ⊢ (φ → ∀yφ)
9	8	eumo0 1022	. . . 4 ⊢ (∃!xφ → ∃y∀x(φ → x = y))
10	7, 9	ja 118	. . 3 ⊢ ((∃xφ → ∃!xφ) → ∃y∀x(φ → x = y))
11	8	eu3 1024	. . . . . 6 ⊢ (∃!xφ ↔ (∃xφ ∧ ∃y∀x(φ → x = y)))
12	11	biimpr 134	. . . . 5 ⊢ ((∃xφ ∧ ∃y∀x(φ → x = y)) → ∃!xφ)
13	12	exp 291	. . . 4 ⊢ (∃xφ → (∃y∀x(φ → x = y) → ∃!xφ))
14	13	com12 13	. . 3 ⊢ (∃y∀x(φ → x = y) → (∃xφ → ∃!xφ))
15	10, 14	impbi 139	. 2 ⊢ ((∃xφ → ∃!xφ) ↔ ∃y∀x(φ → x = y))
16	1, 15	bitr 151	1 ⊢ (∃*xφ ↔ ∃y∀x(φ → x = y))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797  ∃!weu 1007  ∃*wmo 1008

This theorem is referenced by:  mo3 1027  eu5 1035  immo 1043  moimv 1044  moanim 1051  mo2icl 1434  moabex 1868  dffun3 2675  dffunmof 2678

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010

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