Theorem euf 2019

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 2017	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧))
2		euf.1	. . . . 5 ⊢ (𝜑 → ∀𝑦𝜑)
3		ax-17 1514	. . . . 5 ⊢ (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧)
4	2, 3	hbbi 1536	. . . 4 ⊢ ((𝜑 ↔ 𝑥 = 𝑧) → ∀𝑦(𝜑 ↔ 𝑥 = 𝑧))
5	4	hbal 1465	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∀𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑧))
6		ax-17 1514	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∀𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
7		equequ2 1701	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦))
8	7	bibi2d 231	. . . 4 ⊢ (𝑧 = 𝑦 → ((𝜑 ↔ 𝑥 = 𝑧) ↔ (𝜑 ↔ 𝑥 = 𝑦)))
9	8	albidv 1812	. . 3 ⊢ (𝑧 = 𝑦 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
10	5, 6, 9	cbvexh 1743	. 2 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
11	1, 10	bitri 183	1 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1341  ∃wex 1480  ∃!weu 2014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-eu 2017

This theorem is referenced by:  eu1  2039  eumo0  2045