Theorem mobid 2489

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 2091	. . 3 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
4	1, 2	eubid 2488	. . 3 ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))
5	3, 4	imbi12d 334	. 2 ⊢ (𝜑 → ((∃𝑥𝜓 → ∃!𝑥𝜓) ↔ (∃𝑥𝜒 → ∃!𝑥𝜒)))
6		df-mo 2475	. 2 ⊢ (∃*𝑥𝜓 ↔ (∃𝑥𝜓 → ∃!𝑥𝜓))
7		df-mo 2475	. 2 ⊢ (∃*𝑥𝜒 ↔ (∃𝑥𝜒 → ∃!𝑥𝜒))
8	5, 6, 7	3bitr4g 303	1 ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 196  ∃wex 1704  Ⅎwnf 1708  ∃!weu 2470  ∃*wmo 2471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475

This theorem is referenced by:  mobidv  2491  moanim  2529  rmobida  3129  rmoeq1f  3137  funcnvmptOLD  29467  funcnvmpt  29468