Theorem mobid 2238

Description: Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by NM, 8-Mar-1995.)

Hypotheses

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

Assertion

Ref	Expression
mobid	⊢ (φ → (∃*xψ ↔ ∃*xχ))

Proof of Theorem mobid

Step	Hyp	Ref	Expression
1		mobid.1	. . . 4 ⊢ Ⅎxφ
2		mobid.2	. . . 4 ⊢ (φ → (ψ ↔ χ))
3	1, 2	exbid 1773	. . 3 ⊢ (φ → (∃xψ ↔ ∃xχ))
4	1, 2	eubid 2211	. . 3 ⊢ (φ → (∃!xψ ↔ ∃!xχ))
5	3, 4	imbi12d 311	. 2 ⊢ (φ → ((∃xψ → ∃!xψ) ↔ (∃xχ → ∃!xχ)))
6		df-mo 2209	. 2 ⊢ (∃*xψ ↔ (∃xψ → ∃!xψ))
7		df-mo 2209	. 2 ⊢ (∃*xχ ↔ (∃xχ → ∃!xχ))
8	5, 6, 7	3bitr4g 279	1 ⊢ (φ → (∃*xψ ↔ ∃*xχ))

Syntax hints:   → wi 4   ↔ wb 176  ∃wex 1541  Ⅎwnf 1544  ∃!weu 2204  ∃*wmo 2205

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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  df-mo 2209

This theorem is referenced by:  mobidv  2239  euan  2261  rmobida  2798  rmoeq1f  2806