Theorem eubid 2582

Description: Formula-building rule for the unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.) (Proof shortened by Wolf Lammen, 19-Feb-2023.)

Hypotheses

Ref	Expression
eubid.1	⊢ Ⅎ𝑥𝜑
eubid.2	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
eubid	⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))

Proof of Theorem eubid

Step	Hyp	Ref	Expression
1		eubid.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		eubid.2	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	1, 2	alrimi 2201	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒))
4		eubi 2579	. 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))
5	3, 4	syl 17	1 ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1535  Ⅎwnf 1781  ∃!weu 2563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-12 2166

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1778  df-nf 1782  df-mo 2535  df-eu 2564

This theorem is referenced by:  euor  2608  euor2  2610  euan  2618  reubida  3323  reueq1f  3336  eusv2i  5320  reusv2lem3  5326  eubiOLD  42078