Theorem eubid 2589

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 2210	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒))
4		eubi 2586	. 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))
5	3, 4	syl 17	1 ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1540  Ⅎwnf 1790  ∃!weu 2570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-12 2175

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1787  df-nf 1791  df-mo 2542  df-eu 2571

This theorem is referenced by:  euor  2615  euor2  2617  euan  2625  reubida  3320  reueq1f  3333  eusv2i  5321  reusv2lem3  5327  eubiOLD  42022