Theorem euf 1905

Description: A version of the existential uniqueness definition with a hypothesis instead of a distinct variable condition. (Contributed by NM, 12-Aug-1993.)

Hypothesis

Ref	Expression
euf.1	⊢ (𝜑 → ∀𝑦𝜑)

Assertion

Ref	Expression
euf	⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem euf

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-eu 1903	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧))
2		euf.1	. . . . 5 ⊢ (𝜑 → ∀𝑦𝜑)
3		ax-17 1419	. . . . 5 ⊢ (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧)
4	2, 3	hbbi 1440	. . . 4 ⊢ ((𝜑 ↔ 𝑥 = 𝑧) → ∀𝑦(𝜑 ↔ 𝑥 = 𝑧))
5	4	hbal 1366	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∀𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑧))
6		ax-17 1419	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∀𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
7		equequ2 1599	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦))
8	7	bibi2d 221	. . . 4 ⊢ (𝑧 = 𝑦 → ((𝜑 ↔ 𝑥 = 𝑧) ↔ (𝜑 ↔ 𝑥 = 𝑦)))
9	8	albidv 1705	. . 3 ⊢ (𝑧 = 𝑦 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
10	5, 6, 9	cbvexh 1638	. 2 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
11	1, 10	bitri 173	1 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))

Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241  ∃wex 1381  ∃!weu 1900

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-eu 1903

This theorem is referenced by:  eu1  1925  eumo0  1931