Theorem axprlem4 5399

Description: Lemma for axpr 5400. 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 1841	. . . . . . 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 1862	. . 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 1947	. 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 1078  ∀wal 1568  ∃wex 1809  ∀wral 3079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-ral 3080

This theorem is referenced by:  axpr  5400