Theorem axprlem4 5388

Description: Lemma for axpr 5389. If an existing set of empty sets corresponds to one element of the pair, then the element is included in any superset of the set whose existence is asserted by the axiom of replacement. (Contributed by Rohan Ridenour, 10-Aug-2023.) (Revised by BJ, 13-Aug-2023.) (Revised by Matthew House, 18-Sep-2025.)

Hypotheses

Ref	Expression
axprlem4.1	⊢ ∃𝑠∀𝑛𝜑
axprlem4.2	⊢ (𝜑 → (𝑛 ∈ 𝑠 → ∀𝑡 ¬ 𝑡 ∈ 𝑛))
axprlem4.3	⊢ (∀𝑛𝜑 → (if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) ↔ 𝑤 = 𝑣))

Assertion

Ref	Expression
axprlem4	⊢ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) → (𝑤 = 𝑣 → ∃𝑠(𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))))

Distinct variable groups:   𝑤,𝑠   𝑣,𝑠

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑤,𝑣,𝑡,𝑛,𝑠,𝑝)

Proof of Theorem axprlem4

Step	Hyp	Ref	Expression
1		axprlem4.1	. . 3 ⊢ ∃𝑠∀𝑛𝜑
2		axprlem4.2	. . . . . . . 8 ⊢ (𝜑 → (𝑛 ∈ 𝑠 → ∀𝑡 ¬ 𝑡 ∈ 𝑛))
3	2	alimi 1834	. . . . . . 7 ⊢ (∀𝑛𝜑 → ∀𝑛(𝑛 ∈ 𝑠 → ∀𝑡 ¬ 𝑡 ∈ 𝑛))
4	3	ralrid 3087	. . . . . 6 ⊢ (∀𝑛𝜑 → ∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛)
5	4	imim1i 64	. . . . 5 ⊢ ((∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) → (∀𝑛𝜑 → 𝑠 ∈ 𝑝))
6	5	ancrd 560	. . . 4 ⊢ ((∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) → (∀𝑛𝜑 → (𝑠 ∈ 𝑝 ∧ ∀𝑛𝜑)))
7	6	aleximi 1855	. . 3 ⊢ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) → (∃𝑠∀𝑛𝜑 → ∃𝑠(𝑠 ∈ 𝑝 ∧ ∀𝑛𝜑)))
8	1, 7	mpi 21	. 2 ⊢ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) → ∃𝑠(𝑠 ∈ 𝑝 ∧ ∀𝑛𝜑))
9		axprlem4.3	. . . . 5 ⊢ (∀𝑛𝜑 → (if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) ↔ 𝑤 = 𝑣))
10	9	biimprcd 253	. . . 4 ⊢ (𝑤 = 𝑣 → (∀𝑛𝜑 → if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))
11	10	anim2d 623	. . 3 ⊢ (𝑤 = 𝑣 → ((𝑠 ∈ 𝑝 ∧ ∀𝑛𝜑) → (𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))))
12	11	eximdv 1940	. 2 ⊢ (𝑤 = 𝑣 → (∃𝑠(𝑠 ∈ 𝑝 ∧ ∀𝑛𝜑) → ∃𝑠(𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))))
13	8, 12	syl5com 32	1 ⊢ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) → (𝑤 = 𝑣 → ∃𝑠(𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400  if-wif 1076  ∀wal 1561  ∃wex 1802  ∀wral 3079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1803  df-ral 3080

This theorem is referenced by:  axpr  5389