Theorem eumo0 1980

Description: Existential uniqueness implies "at most one." (Contributed by NM, 8-Jul-1994.)

Hypothesis

Ref	Expression
eumo0.1	⊢ (𝜑 → ∀𝑦𝜑)

Assertion

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

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem eumo0

Step	Hyp	Ref	Expression
1		eumo0.1	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
2	1	euf 1954	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
3		bi1 117	. . . 4 ⊢ ((𝜑 ↔ 𝑥 = 𝑦) → (𝜑 → 𝑥 = 𝑦))
4	3	alimi 1390	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∀𝑥(𝜑 → 𝑥 = 𝑦))
5	4	eximi 1537	. 2 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
6	2, 5	sylbi 120	1 ⊢ (∃!𝑥𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1288  ∃wex 1427  ∃!weu 1949

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474

This theorem depends on definitions:  df-bi 116  df-eu 1952

This theorem is referenced by:  eu2  1993  eu3h  1994