Theorem mo2 1399

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:   x,y

Proof of Theorem mo2

Step	Hyp	Ref	Expression
1		df-mo 1382	. 2 ⊢ (∃*xφ ↔ (∃xφ → ∃!xφ))
2		alnex 1032	. . . . 5 ⊢ (∀x ¬ φ ↔ ¬ ∃xφ)
3		pm2.21 76	. . . . . . 7 ⊢ (¬ φ → (φ → x = y))
4	3	19.20i 991	. . . . . 6 ⊢ (∀x ¬ φ → ∀x(φ → x = y))
5		19.8a 1028	. . . . . 6 ⊢ (∀x(φ → x = y) → ∃y∀x(φ → x = y))
6	4, 5	syl 10	. . . . 5 ⊢ (∀x ¬ φ → ∃y∀x(φ → x = y))
7	2, 6	sylbir 201	. . . 4 ⊢ (¬ ∃xφ → ∃y∀x(φ → x = y))
8		mo2.1	. . . . 5 ⊢ (φ → ∀yφ)
9	8	eumo0 1394	. . . 4 ⊢ (∃!xφ → ∃y∀x(φ → x = y))
10	7, 9	ja 137	. . 3 ⊢ ((∃xφ → ∃!xφ) → ∃y∀x(φ → x = y))
11	8	eu3 1396	. . . . 5 ⊢ (∃!xφ ↔ (∃xφ ⋀ ∃y∀x(φ → x = y)))
12	11	biimpr 152	. . . 4 ⊢ ((∃xφ ⋀ ∃y∀x(φ → x = y)) → ∃!xφ)
13	12	expcom 374	. . 3 ⊢ (∃y∀x(φ → x = y) → (∃xφ → ∃!xφ))
14	10, 13	impbi 157	. 2 ⊢ ((∃xφ → ∃!xφ) ↔ ∃y∀x(φ → x = y))
15	1, 14	bitr 173	1 ⊢ (∃*xφ ↔ ∃y∀x(φ → x = y))

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146   ⋀ wa 223  ∀wal 953   = wceq 955  ∃wex 979  ∃!weu 1379  ∃*wmo 1380

This theorem is referenced by:  mo3 1400  eu5 1408  immo 1416  moimv 1418  moanim 1426  mo2icl 1920  moabex 2762  dffun3 3523  dffunmof 3526  grothprim 8738

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-8 963  ax-10 965  ax-11 966  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382

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