Theorem mobi 2573

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 1922	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜓 → 𝜑)))
2		moim 2570	. . . 4 ⊢ (∀𝑥(𝜓 → 𝜑) → (∃*𝑥𝜑 → ∃*𝑥𝜓))
3		moim 2570	. . . 4 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑))
4	2, 3	impbid21d 214	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥(𝜓 → 𝜑) → (∃*𝑥𝜑 ↔ ∃*𝑥𝜓)))
5	4	imp 412	. 2 ⊢ ((∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜓 → 𝜑)) → (∃*𝑥𝜑 ↔ ∃*𝑥𝜓))
6	1, 5	sylbi 220	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃*𝑥𝜑 ↔ ∃*𝑥𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568  ∃*wmo 2563

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-mo 2565

This theorem is used by:  mobii  2574  mobidv  2575  mobid  2576  eubi  2610