Theorem eubid 1378

Description: Formula-building rule for uniqueness quantifier (deduction rule).

Hypotheses

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

Assertion

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

Proof of Theorem eubid

Step	Hyp	Ref	Expression
1		eubid.1	. . . 4 ⊢ (φ → ∀xφ)
2		eubid.2	. . . . 5 ⊢ (φ → (ψ ↔ χ))
3	2	bibi1d 617	. . . 4 ⊢ (φ → ((ψ ↔ x = y) ↔ (χ ↔ x = y)))
4	1, 3	albid 1100	. . 3 ⊢ (φ → (∀x(ψ ↔ x = y) ↔ ∀x(χ ↔ x = y)))
5	4	exbidv 1274	. 2 ⊢ (φ → (∃y∀x(ψ ↔ x = y) ↔ ∃y∀x(χ ↔ x = y)))
6		df-eu 1375	. 2 ⊢ (∃!xψ ↔ ∃y∀x(ψ ↔ x = y))
7		df-eu 1375	. 2 ⊢ (∃!xχ ↔ ∃y∀x(χ ↔ x = y))
8	5, 6, 7	3bitr4g 553	1 ⊢ (φ → (∃!xψ ↔ ∃!xχ))

Syntax hints:   → wi 3   ↔ wb 146  ∀wal 951  ∃wex 977  ∃!weu 1373

This theorem is referenced by:  eubidv 1379  eubii 1380  euor 1391  mobid 1397  reueq1f 1777

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-17 968  ax-4 970  ax-5o 972

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-eu 1375

Copyright terms: Public domain