Theorem mobi 2576

Description: Equivalence theorem for the at-most-one quantifier. (Contributed by BJ, 7-Oct-2022.) (Proof shortened by Wolf Lammen, 18-Feb-2023.)

Assertion

Ref	Expression
mobi	⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃*𝑥𝜑 ↔ ∃*𝑥𝜓))

Proof of Theorem mobi

Step	Hyp	Ref	Expression
1		albiim 1911	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜓 → 𝜑)))
2		moim 2573	. . . 4 ⊢ (∀𝑥(𝜓 → 𝜑) → (∃*𝑥𝜑 → ∃*𝑥𝜓))
3		moim 2573	. . . 4 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑))
4	2, 3	impbid21d 213	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥(𝜓 → 𝜑) → (∃*𝑥𝜑 ↔ ∃*𝑥𝜓)))
5	4	imp 410	. 2 ⊢ ((∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜓 → 𝜑)) → (∃*𝑥𝜑 ↔ ∃*𝑥𝜓))
6	1, 5	sylbi 219	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃*𝑥𝜑 ↔ ∃*𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1560  ∃*wmo 2566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1802  df-mo 2568

This theorem is referenced by:  mobii  2577  mobidv  2578  mobid  2579  eubi  2613