Theorem mobidh 1975

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

Hypotheses

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

Assertion

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

Proof of Theorem mobidh

Step	Hyp	Ref	Expression
1		mobidh.1	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
2		mobidh.2	. . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	1, 2	exbidh 1545	. . 3 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
4	1, 2	eubidh 1947	. . 3 ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))
5	3, 4	imbi12d 232	. 2 ⊢ (𝜑 → ((∃𝑥𝜓 → ∃!𝑥𝜓) ↔ (∃𝑥𝜒 → ∃!𝑥𝜒)))
6		df-mo 1945	. 2 ⊢ (∃*𝑥𝜓 ↔ (∃𝑥𝜓 → ∃!𝑥𝜓))
7		df-mo 1945	. 2 ⊢ (∃*𝑥𝜒 ↔ (∃𝑥𝜒 → ∃!𝑥𝜒))
8	5, 6, 7	3bitr4g 221	1 ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 103  ∀wal 1282  ∃wex 1421  ∃!weu 1941  ∃*wmo 1942

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  df-mo 1945

This theorem is referenced by:  euan  1997