Theorem mobid 2578

Description: Formula-building rule for the at-most-one quantifier (deduction form). (Contributed by NM, 8-Mar-1995.) Remove dependency on ax-10 2176, ax-11 2192, ax-13 2404. (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 2249	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒))
4		mobi 2575	. 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
5	3, 4	syl 18	1 ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1568  Ⅎwnf 1813  ∃*wmo 2565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-12 2213

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-nf 1814  df-mo 2567

This theorem is referenced by:  moanim  2648  rmobida  3392  rmoeq1f  3406  funcnvmpt  6991