Theorem eubid 2052

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

Hypotheses

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

Assertion

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

Proof of Theorem eubid

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eubid.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2		eubid.2	. . . . 5 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	2	bibi1d 233	. . . 4 ⊢ (𝜑 → ((𝜓 ↔ 𝑥 = 𝑦) ↔ (𝜒 ↔ 𝑥 = 𝑦)))
4	1, 3	albid 1629	. . 3 ⊢ (𝜑 → (∀𝑥(𝜓 ↔ 𝑥 = 𝑦) ↔ ∀𝑥(𝜒 ↔ 𝑥 = 𝑦)))
5	4	exbidv 1839	. 2 ⊢ (𝜑 → (∃𝑦∀𝑥(𝜓 ↔ 𝑥 = 𝑦) ↔ ∃𝑦∀𝑥(𝜒 ↔ 𝑥 = 𝑦)))
6		df-eu 2048	. 2 ⊢ (∃!𝑥𝜓 ↔ ∃𝑦∀𝑥(𝜓 ↔ 𝑥 = 𝑦))
7		df-eu 2048	. 2 ⊢ (∃!𝑥𝜒 ↔ ∃𝑦∀𝑥(𝜒 ↔ 𝑥 = 𝑦))
8	5, 6, 7	3bitr4g 223	1 ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1362  Ⅎwnf 1474  ∃wex 1506  ∃!weu 2045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-eu 2048

This theorem is referenced by:  eubidv  2053  mobid  2080  reubida  2679  reueq1f  2691  eusv2i  4490