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Metamath Proof Explorer

Theorem mobid

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

	Ref	Expression
Hypotheses	mobid.1	⊢ Ⅎ 𝑥 𝜑
	mobid.2	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
Assertion	mobid	⊢ ( 𝜑 → ( ∃* 𝑥 𝜓 ↔ ∃* 𝑥 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		mobid.1	⊢ Ⅎ 𝑥 𝜑
2		mobid.2	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
3	1 2	alrimi	⊢ ( 𝜑 → ∀ 𝑥 ( 𝜓 ↔ 𝜒 ) )
4		mobi	⊢ ( ∀ 𝑥 ( 𝜓 ↔ 𝜒 ) → ( ∃* 𝑥 𝜓 ↔ ∃* 𝑥 𝜒 ) )
5	3 4	syl	⊢ ( 𝜑 → ( ∃* 𝑥 𝜓 ↔ ∃* 𝑥 𝜒 ) )