Theorem eubid 2587

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 2213	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒))
4		eubi 2584	. 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))
5	3, 4	syl 17	1 ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538  Ⅎwnf 1783  ∃!weu 2568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-12 2177

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784  df-mo 2540  df-eu 2569

This theorem is referenced by:  euor  2611  euor2  2613  euan  2621  reubida  3407  eusv2i  5394  reusv2lem3  5400  eubiOLD  44455