Theorem mobidh 2053

Description: Formula-building rule for "at most one" quantifier (deduction form). (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 1607	. . 3 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
4	1, 2	eubidh 2025	. . 3 ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))
5	3, 4	imbi12d 233	. 2 ⊢ (𝜑 → ((∃𝑥𝜓 → ∃!𝑥𝜓) ↔ (∃𝑥𝜒 → ∃!𝑥𝜒)))
6		df-mo 2023	. 2 ⊢ (∃*𝑥𝜓 ↔ (∃𝑥𝜓 → ∃!𝑥𝜓))
7		df-mo 2023	. 2 ⊢ (∃*𝑥𝜒 ↔ (∃𝑥𝜒 → ∃!𝑥𝜒))
8	5, 6, 7	3bitr4g 222	1 ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1346  ∃wex 1485  ∃!weu 2019  ∃*wmo 2020

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-eu 2022  df-mo 2023

This theorem is referenced by:  euan  2075