Theorem euanv 2651

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 2604	. . . 4 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓))
2		simpl 486	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
3	2	exlimiv 1950	. . . 4 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → 𝜑)
4	1, 3	syl 17	. . 3 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → 𝜑)
5		ibar 536	. . . . 5 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜓)))
6	5	eubidv 2613	. . . 4 ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥(𝜑 ∧ 𝜓)))
7	6	biimprcd 252	. . 3 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → (𝜑 → ∃!𝑥𝜓))
8	4, 7	jcai 524	. 2 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → (𝜑 ∧ ∃!𝑥𝜓))
9	6	biimpa 480	. 2 ⊢ ((𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑 ∧ 𝜓))
10	8, 9	impbii 211	1 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓))

Syntax hints:   ↔ wb 208   ∧ wa 399  ∃wex 1799  ∃!weu 2595

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1800  df-mo 2566  df-eu 2596

This theorem is referenced by:  eueq2  3673  2reu5lem1  3718  fsn  7117  dfac5lem5  10083