Theorem axprlem4 5387

Description: Lemma for axpr 5388. 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 1843	. . . . . . 7 ⊢ (∀𝑛𝜑 → ∀𝑛(𝑛 ∈ 𝑠 → ∀𝑡 ¬ 𝑡 ∈ 𝑛))
4	3	ralrid 3084	. . . . . 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 561	. . . 4 ⊢ ((∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) → (∀𝑛𝜑 → (𝑠 ∈ 𝑝 ∧ ∀𝑛𝜑)))
7	6	aleximi 1864	. . 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 624	. . 3 ⊢ (𝑤 = 𝑣 → ((𝑠 ∈ 𝑝 ∧ ∀𝑛𝜑) → (𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))))
12	11	eximdv 1949	. 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))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  if-wif 1078  ∀wal 1567  ∃wex 1811  ∀wral 3076

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-ral 3077

This theorem is used by:  axpr  5388