Theorem euanv 2652

Description: Introduction of a conjunct into unique existential quantifier. (Contributed by NM, 23-Mar-1995.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 14-Jan-2023.)

Assertion

Ref	Expression
euanv	⊢ (∃!𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓))

Distinct variable group:   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem euanv

Step	Hyp	Ref	Expression
1		euex 2605	. . . 4 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓))
2		simpl 488	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
3	2	exlimiv 1964	. . . 4 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → 𝜑)
4	1, 3	syl 18	. . 3 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → 𝜑)
5		ibar 538	. . . . 5 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜓)))
6	5	eubidv 2614	. . . 4 ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥(𝜑 ∧ 𝜓)))
7	6	biimprcd 253	. . 3 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → (𝜑 → ∃!𝑥𝜓))
8	4, 7	jcai 526	. 2 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → (𝜑 ∧ ∃!𝑥𝜓))
9	6	biimpa 482	. 2 ⊢ ((𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑 ∧ 𝜓))
10	8, 9	impbii 212	1 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401  ∃wex 1811  ∃!weu 2596

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-mo 2567  df-eu 2597

This theorem is used by:  eueq2  3671  2reu5lem1  3716  fsn  7147  dfac5lem5  10219