Theorem opnneilv 49822

Description: The converse of opnneir 49820 with different dummy variables. Note that the second hypothesis could be generalized by adding 𝑦 ∈ 𝐽 to the antecedent. See the proof for details. Although 𝐽 ∈ Top might be redundant here (see neircl 49818), it is listed for explicitness. (Contributed by Zhi Wang, 31-Aug-2024.)

Hypotheses

Ref	Expression
opnneir.1	⊢ (𝜑 → 𝐽 ∈ Top)
opnneilv.2	⊢ ((𝜑 ∧ 𝑦 ⊆ 𝑥) → (𝜓 → 𝜒))

Assertion

Ref	Expression
opnneilv	⊢ (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 → ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒)))

Distinct variable groups:   𝑥,𝐽,𝑦   𝑥,𝑆,𝑦   𝜒,𝑥   𝜑,𝑥,𝑦   𝜓,𝑦

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)

Proof of Theorem opnneilv

Step	Hyp	Ref	Expression
1		df-rex 3089	. 2 ⊢ (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 ↔ ∃𝑥(𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝜓))
2		opnneir.1	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ Top)
3		neii2 23334	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑥))
4	2, 3	sylan 592	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑥))
5	4	r19.41dv 49717	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝜓) → ∃𝑦 ∈ 𝐽 ((𝑆 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑥) ∧ 𝜓))
6	5	expl 463	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝜓) → ∃𝑦 ∈ 𝐽 ((𝑆 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑥) ∧ 𝜓)))
7		anass 474	. . . . . 6 ⊢ (((𝑆 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑥) ∧ 𝜓) ↔ (𝑆 ⊆ 𝑦 ∧ (𝑦 ⊆ 𝑥 ∧ 𝜓)))
8		opnneilv.2	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝑥) → (𝜓 → 𝜒))
9	8	expimpd 459	. . . . . . 7 ⊢ (𝜑 → ((𝑦 ⊆ 𝑥 ∧ 𝜓) → 𝜒))
10	9	anim2d 624	. . . . . 6 ⊢ (𝜑 → ((𝑆 ⊆ 𝑦 ∧ (𝑦 ⊆ 𝑥 ∧ 𝜓)) → (𝑆 ⊆ 𝑦 ∧ 𝜒)))
11	7, 10	biimtrid 245	. . . . 5 ⊢ (𝜑 → (((𝑆 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑥) ∧ 𝜓) → (𝑆 ⊆ 𝑦 ∧ 𝜒)))
12	11	reximdv 3179	. . . 4 ⊢ (𝜑 → (∃𝑦 ∈ 𝐽 ((𝑆 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑥) ∧ 𝜓) → ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒)))
13	6, 12	syld 48	. . 3 ⊢ (𝜑 → ((𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝜓) → ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒)))
14	13	exlimdv 1966	. 2 ⊢ (𝜑 → (∃𝑥(𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝜓) → ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒)))
15	1, 14	biimtrid 245	1 ⊢ (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 → ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∃wex 1812   ∈ wcel 2145  ∃wrex 3088   ⊆ wss 3902  ‘cfv 6537  Topctop 23119  neicnei 23323

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-top 23120  df-nei 23324

This theorem is used by:  opnneil  49823