Theorem mobid 2548

Description: Formula-building rule for the at-most-one quantifier (deduction form). (Contributed by NM, 8-Mar-1995.) Remove dependency on ax-10 2139, ax-11 2155, ax-13 2375. (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 2211	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒))
4		mobi 2545	. 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
5	3, 4	syl 17	1 ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1535  Ⅎwnf 1780  ∃*wmo 2536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-12 2175

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-nf 1781  df-mo 2538

This theorem is referenced by:  moanim  2618  rmobida  3404  rmoeq1f  3421  funcnvmpt  32684