Theorem mosub

Description: "At most one" remains true after substitution. (Contributed by NM, 9-Mar-1995)

	Ref	Expression
Hypothesis	mosub.1	⊢ ∃* 𝑥 𝜑
Assertion	mosub	⊢ ∃* 𝑥 ∃ 𝑦 ( 𝑦 = 𝐴 ∧ 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		mosub.1	⊢ ∃* 𝑥 𝜑
2		moeq	⊢ ∃* 𝑦 𝑦 = 𝐴
3	1	ax-gen	⊢ ∀ 𝑦 ∃* 𝑥 𝜑
4		moexexvw	⊢ ( ( ∃* 𝑦 𝑦 = 𝐴 ∧ ∀ 𝑦 ∃* 𝑥 𝜑 ) → ∃* 𝑥 ∃ 𝑦 ( 𝑦 = 𝐴 ∧ 𝜑 ) )
5	2 3 4	mp2an	⊢ ∃* 𝑥 ∃ 𝑦 ( 𝑦 = 𝐴 ∧ 𝜑 )