Theorem rmoan

Description: Restricted "at most one" still holds when a conjunct is added. (Contributed by NM, 16-Jun-2017)

	Ref	Expression
Assertion	rmoan	⊢ ( ∃* 𝑥 ∈ 𝐴 𝜑 → ∃* 𝑥 ∈ 𝐴 ( 𝜓 ∧ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		moan	⊢ ( ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → ∃* 𝑥 ( 𝜓 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) )
2		an12	⊢ ( ( 𝜓 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) ↔ ( 𝑥 ∈ 𝐴 ∧ ( 𝜓 ∧ 𝜑 ) ) )
3	2	mobii	⊢ ( ∃* 𝑥 ( 𝜓 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) ↔ ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ ( 𝜓 ∧ 𝜑 ) ) )
4	1 3	sylib	⊢ ( ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ ( 𝜓 ∧ 𝜑 ) ) )
5		df-rmo	⊢ ( ∃* 𝑥 ∈ 𝐴 𝜑 ↔ ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
6		df-rmo	⊢ ( ∃* 𝑥 ∈ 𝐴 ( 𝜓 ∧ 𝜑 ) ↔ ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ ( 𝜓 ∧ 𝜑 ) ) )
7	4 5 6	3imtr4i	⊢ ( ∃* 𝑥 ∈ 𝐴 𝜑 → ∃* 𝑥 ∈ 𝐴 ( 𝜓 ∧ 𝜑 ) )