Theorem eubid 2211

Description: Formula-building rule for uniqueness quantifier (deduction rule). (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 310	. . . 4 ⊢ (φ → ((ψ ↔ x = y) ↔ (χ ↔ x = y)))
4	1, 3	albid 1772	. . 3 ⊢ (φ → (∀x(ψ ↔ x = y) ↔ ∀x(χ ↔ x = y)))
5	4	exbidv 1626	. 2 ⊢ (φ → (∃y∀x(ψ ↔ x = y) ↔ ∃y∀x(χ ↔ x = y)))
6		df-eu 2208	. 2 ⊢ (∃!xψ ↔ ∃y∀x(ψ ↔ x = y))
7		df-eu 2208	. 2 ⊢ (∃!xχ ↔ ∃y∀x(χ ↔ x = y))
8	5, 6, 7	3bitr4g 279	1 ⊢ (φ → (∃!xψ ↔ ∃!xχ))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544   = wceq 1642  ∃!weu 2204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545  df-eu 2208

This theorem is referenced by:  eubidv  2212  euor  2231  mobid  2238  euan  2261  eupickbi  2270  euor2  2272  reubida  2794  reueq1f  2806