Metamath Proof Explorer


Theorem opnneilv

Description: The converse of opnneir with different dummy variables. Note that the second hypothesis could be generalized by adding y e. J to the antecedent. See the proof for details. Although J e. Top might be redundant here (see neircl ), it is listed for explicitness. (Contributed by Zhi Wang, 31-Aug-2024)

Ref Expression
Hypotheses opnneir.1 ( 𝜑𝐽 ∈ Top )
opnneilv.2 ( ( 𝜑𝑦𝑥 ) → ( 𝜓𝜒 ) )
Assertion opnneilv ( 𝜑 → ( ∃ 𝑥 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑆 ) 𝜓 → ∃ 𝑦𝐽 ( 𝑆𝑦𝜒 ) ) )

Proof

Step Hyp Ref Expression
1 opnneir.1 ( 𝜑𝐽 ∈ Top )
2 opnneilv.2 ( ( 𝜑𝑦𝑥 ) → ( 𝜓𝜒 ) )
3 df-rex ( ∃ 𝑥 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑆 ) 𝜓 ↔ ∃ 𝑥 ( 𝑥 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑆 ) ∧ 𝜓 ) )
4 neii2 ( ( 𝐽 ∈ Top ∧ 𝑥 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑆 ) ) → ∃ 𝑦𝐽 ( 𝑆𝑦𝑦𝑥 ) )
5 1 4 sylan ( ( 𝜑𝑥 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑆 ) ) → ∃ 𝑦𝐽 ( 𝑆𝑦𝑦𝑥 ) )
6 5 r19.41dv ( ( ( 𝜑𝑥 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑆 ) ) ∧ 𝜓 ) → ∃ 𝑦𝐽 ( ( 𝑆𝑦𝑦𝑥 ) ∧ 𝜓 ) )
7 6 expl ( 𝜑 → ( ( 𝑥 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑆 ) ∧ 𝜓 ) → ∃ 𝑦𝐽 ( ( 𝑆𝑦𝑦𝑥 ) ∧ 𝜓 ) ) )
8 anass ( ( ( 𝑆𝑦𝑦𝑥 ) ∧ 𝜓 ) ↔ ( 𝑆𝑦 ∧ ( 𝑦𝑥𝜓 ) ) )
9 2 expimpd ( 𝜑 → ( ( 𝑦𝑥𝜓 ) → 𝜒 ) )
10 9 anim2d ( 𝜑 → ( ( 𝑆𝑦 ∧ ( 𝑦𝑥𝜓 ) ) → ( 𝑆𝑦𝜒 ) ) )
11 8 10 syl5bi ( 𝜑 → ( ( ( 𝑆𝑦𝑦𝑥 ) ∧ 𝜓 ) → ( 𝑆𝑦𝜒 ) ) )
12 11 reximdv ( 𝜑 → ( ∃ 𝑦𝐽 ( ( 𝑆𝑦𝑦𝑥 ) ∧ 𝜓 ) → ∃ 𝑦𝐽 ( 𝑆𝑦𝜒 ) ) )
13 7 12 syld ( 𝜑 → ( ( 𝑥 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑆 ) ∧ 𝜓 ) → ∃ 𝑦𝐽 ( 𝑆𝑦𝜒 ) ) )
14 13 exlimdv ( 𝜑 → ( ∃ 𝑥 ( 𝑥 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑆 ) ∧ 𝜓 ) → ∃ 𝑦𝐽 ( 𝑆𝑦𝜒 ) ) )
15 3 14 syl5bi ( 𝜑 → ( ∃ 𝑥 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑆 ) 𝜓 → ∃ 𝑦𝐽 ( 𝑆𝑦𝜒 ) ) )