Theorem eubidh 1947

Description: Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.)

Hypotheses

Ref	Expression
eubidh.1	⊢ (𝜑 → ∀𝑥𝜑)
eubidh.2	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

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

Proof of Theorem eubidh

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eubidh.1	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
2		eubidh.2	. . . . 5 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	2	bibi1d 231	. . . 4 ⊢ (𝜑 → ((𝜓 ↔ 𝑥 = 𝑦) ↔ (𝜒 ↔ 𝑥 = 𝑦)))
4	1, 3	albidh 1409	. . 3 ⊢ (𝜑 → (∀𝑥(𝜓 ↔ 𝑥 = 𝑦) ↔ ∀𝑥(𝜒 ↔ 𝑥 = 𝑦)))
5	4	exbidv 1746	. 2 ⊢ (𝜑 → (∃𝑦∀𝑥(𝜓 ↔ 𝑥 = 𝑦) ↔ ∃𝑦∀𝑥(𝜒 ↔ 𝑥 = 𝑦)))
6		df-eu 1944	. 2 ⊢ (∃!𝑥𝜓 ↔ ∃𝑦∀𝑥(𝜓 ↔ 𝑥 = 𝑦))
7		df-eu 1944	. 2 ⊢ (∃!𝑥𝜒 ↔ ∃𝑦∀𝑥(𝜒 ↔ 𝑥 = 𝑦))
8	5, 6, 7	3bitr4g 221	1 ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 103  ∀wal 1282  ∃wex 1421  ∃!weu 1941

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-eu 1944

This theorem is referenced by:  euor  1967  mobidh  1975  euan  1997  euor2  1999  eupickbi  2023