Theorem euanv 2703

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 2656	. . . 4 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓))
2		simpl 485	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
3	2	exlimiv 1925	. . . 4 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → 𝜑)
4	1, 3	syl 17	. . 3 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → 𝜑)
5		ibar 531	. . . . 5 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜓)))
6	5	eubidv 2666	. . . 4 ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥(𝜑 ∧ 𝜓)))
7	6	biimprcd 252	. . 3 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → (𝜑 → ∃!𝑥𝜓))
8	4, 7	jcai 519	. 2 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → (𝜑 ∧ ∃!𝑥𝜓))
9	6	biimpa 479	. 2 ⊢ ((𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑 ∧ 𝜓))
10	8, 9	impbii 211	1 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓))

Syntax hints:   ↔ wb 208   ∧ wa 398  ∃wex 1774  ∃!weu 2647

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1775  df-mo 2616  df-eu 2648

This theorem is referenced by:  eueq2  3699  2reu5lem1  3744  fsn  6890  dfac5lem5  9545