Theorem euan 2647

Description: Introduction of a conjunct into unique existential quantifier. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 24-Dec-2018.)

Hypothesis

Ref	Expression
moanim.1	⊢ Ⅎ𝑥𝜑

Assertion

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

Proof of Theorem euan

Step	Hyp	Ref	Expression
1		euex 2603	. . . 4 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓))
2		moanim.1	. . . . 5 ⊢ Ⅎ𝑥𝜑
3		simpl 486	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
4	2, 3	exlimi 2251	. . . 4 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → 𝜑)
5	1, 4	syl 17	. . 3 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → 𝜑)
6		ibar 536	. . . . 5 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜓)))
7	2, 6	eubid 2613	. . . 4 ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥(𝜑 ∧ 𝜓)))
8	7	biimprcd 252	. . 3 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → (𝜑 → ∃!𝑥𝜓))
9	5, 8	jcai 524	. 2 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → (𝜑 ∧ ∃!𝑥𝜓))
10	7	biimpa 480	. 2 ⊢ ((𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑 ∧ 𝜓))
11	9, 10	impbii 211	1 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓))

Syntax hints:   ↔ wb 208   ∧ wa 399  ∃wex 1798  Ⅎwnf 1802  ∃!weu 2594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-12 2211

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1799  df-nf 1803  df-mo 2565  df-eu 2595

This theorem is referenced by:  2eu7  2683  2eu8  2684