Theorem eumo0 1394

Description: Existential uniqueness implies "at most one."

Hypothesis

Ref	Expression
eumo0.1	⊢ (φ → ∀yφ)

Assertion

Ref	Expression
eumo0	⊢ (∃!xφ → ∃y∀x(φ → x = y))

Distinct variable group:   x,y

Proof of Theorem eumo0

Step	Hyp	Ref	Expression
1		eumo0.1	. . 3 ⊢ (φ → ∀yφ)
2	1	euf 1383	. 2 ⊢ (∃!xφ ↔ ∃y∀x(φ ↔ x = y))
3		bi1 148	. . . 4 ⊢ ((φ ↔ x = y) → (φ → x = y))
4	3	19.20i 991	. . 3 ⊢ (∀x(φ ↔ x = y) → ∀x(φ → x = y))
5	4	19.22i 1039	. 2 ⊢ (∃y∀x(φ ↔ x = y) → ∃y∀x(φ → x = y))
6	2, 5	sylbi 199	1 ⊢ (∃!xφ → ∃y∀x(φ → x = y))

Syntax hints:   → wi 3   ↔ wb 146  ∀wal 953   = wceq 955  ∃wex 979  ∃!weu 1379

This theorem is referenced by:  eu2 1395  mo2 1399

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-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122

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

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