Metamath Proof Explorer


Theorem rmounid

Description: A case where an "at most one" restricted existential quantifier for a union is equivalent to such a quantifier for one of the sets. (Contributed by Thierry Arnoux, 27-Nov-2023)

Ref Expression
Hypothesis rmounid.1 ( ( 𝜑𝑥𝐵 ) → ¬ 𝜓 )
Assertion rmounid ( 𝜑 → ( ∃* 𝑥 ∈ ( 𝐴𝐵 ) 𝜓 ↔ ∃* 𝑥𝐴 𝜓 ) )

Proof

Step Hyp Ref Expression
1 rmounid.1 ( ( 𝜑𝑥𝐵 ) → ¬ 𝜓 )
2 1 ex ( 𝜑 → ( 𝑥𝐵 → ¬ 𝜓 ) )
3 2 con2d ( 𝜑 → ( 𝜓 → ¬ 𝑥𝐵 ) )
4 3 imp ( ( 𝜑𝜓 ) → ¬ 𝑥𝐵 )
5 biorf ( ¬ 𝑥𝐵 → ( 𝑥𝐴 ↔ ( 𝑥𝐵𝑥𝐴 ) ) )
6 orcom ( ( 𝑥𝐴𝑥𝐵 ) ↔ ( 𝑥𝐵𝑥𝐴 ) )
7 5 6 bitr4di ( ¬ 𝑥𝐵 → ( 𝑥𝐴 ↔ ( 𝑥𝐴𝑥𝐵 ) ) )
8 4 7 syl ( ( 𝜑𝜓 ) → ( 𝑥𝐴 ↔ ( 𝑥𝐴𝑥𝐵 ) ) )
9 elun ( 𝑥 ∈ ( 𝐴𝐵 ) ↔ ( 𝑥𝐴𝑥𝐵 ) )
10 8 9 bitr4di ( ( 𝜑𝜓 ) → ( 𝑥𝐴𝑥 ∈ ( 𝐴𝐵 ) ) )
11 10 pm5.32da ( 𝜑 → ( ( 𝜓𝑥𝐴 ) ↔ ( 𝜓𝑥 ∈ ( 𝐴𝐵 ) ) ) )
12 11 biancomd ( 𝜑 → ( ( 𝜓𝑥𝐴 ) ↔ ( 𝑥 ∈ ( 𝐴𝐵 ) ∧ 𝜓 ) ) )
13 12 bicomd ( 𝜑 → ( ( 𝑥 ∈ ( 𝐴𝐵 ) ∧ 𝜓 ) ↔ ( 𝜓𝑥𝐴 ) ) )
14 13 biancomd ( 𝜑 → ( ( 𝑥 ∈ ( 𝐴𝐵 ) ∧ 𝜓 ) ↔ ( 𝑥𝐴𝜓 ) ) )
15 14 mobidv ( 𝜑 → ( ∃* 𝑥 ( 𝑥 ∈ ( 𝐴𝐵 ) ∧ 𝜓 ) ↔ ∃* 𝑥 ( 𝑥𝐴𝜓 ) ) )
16 df-rmo ( ∃* 𝑥 ∈ ( 𝐴𝐵 ) 𝜓 ↔ ∃* 𝑥 ( 𝑥 ∈ ( 𝐴𝐵 ) ∧ 𝜓 ) )
17 df-rmo ( ∃* 𝑥𝐴 𝜓 ↔ ∃* 𝑥 ( 𝑥𝐴𝜓 ) )
18 15 16 17 3bitr4g ( 𝜑 → ( ∃* 𝑥 ∈ ( 𝐴𝐵 ) 𝜓 ↔ ∃* 𝑥𝐴 𝜓 ) )