Theorem mobid 2580

Description: Formula-building rule for the at-most-one quantifier (deduction form). (Contributed by NM, 8-Mar-1995.) Remove dependency on ax-10 2179, ax-11 2195, ax-13 2406. (Revised by BJ, 14-Oct-2022.) (Proof shortened by Wolf Lammen, 18-Feb-2023.)

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	. . 3 ⊢ Ⅎ𝑥𝜑
2		mobid.2	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	1, 2	alrimi 2252	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒))
4		mobi 2577	. 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
5	3, 4	syl 18	1 ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568  Ⅎwnf 1816  ∃*wmo 2567

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  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-mo 2569

This theorem is used by:  moanim  2650  rmobida  3394  rmoeq1f  3408  funcnvmpt  6995