Theorem euanv 2521

Description: Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)

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		nfv 1829	. 2 ⊢ Ⅎ𝑥𝜑
2	1	euan 2517	1 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓))

Syntax hints:   ↔ wb 194   ∧ wa 382  ∃!weu 2457

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-12 2033

This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-ex 1695  df-nf 1700  df-eu 2461

This theorem is referenced by:  eueq2  3346  2reu5lem1  3379  fsn  6292  dfac5lem5  8810