Theorem bdayfinbndcbv 28480

Description: Lemma for bdayfinbnd 28483. Change some bound variables. (Contributed by Scott Fenton, 25-Feb-2026.)

Hypotheses

Ref	Expression
bdayfinbndlem.1	⊢ (𝜑 → 𝑁 ∈ ℕ0s)
bdayfinbndlem.2	⊢ (𝜑 → ∀𝑧 ∈ No ((( bday ‘𝑧) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁))))

Assertion

Ref	Expression
bdayfinbndcbv	⊢ (𝜑 → ∀𝑤 ∈ No ((( bday ‘𝑤) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑤) → (𝑤 = 𝑁 ∨ ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s 𝑁))))

Distinct variable group:   𝑥,𝑦,𝑝,𝑎,𝑏,𝑞,𝑧,𝑤,𝑁

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤,𝑞,𝑝,𝑎,𝑏)

Proof of Theorem bdayfinbndcbv

Step	Hyp	Ref	Expression
1		bdayfinbndlem.2	. 2 ⊢ (𝜑 → ∀𝑧 ∈ No ((( bday ‘𝑧) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁))))
2		fveq2 6831	. . . . . 6 ⊢ (𝑧 = 𝑤 → ( bday ‘𝑧) = ( bday ‘𝑤))
3	2	sseq1d 3948	. . . . 5 ⊢ (𝑧 = 𝑤 → (( bday ‘𝑧) ⊆ ( bday ‘𝑁) ↔ ( bday ‘𝑤) ⊆ ( bday ‘𝑁)))
4		breq2 5079	. . . . 5 ⊢ (𝑧 = 𝑤 → ( 0s ≤s 𝑧 ↔ 0s ≤s 𝑤))
5	3, 4	anbi12d 639	. . . 4 ⊢ (𝑧 = 𝑤 → ((( bday ‘𝑧) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑧) ↔ (( bday ‘𝑤) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑤)))
6		eqeq1 2745	. . . . 5 ⊢ (𝑧 = 𝑤 → (𝑧 = 𝑁 ↔ 𝑤 = 𝑁))
7		eqeq1 2745	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ↔ 𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝)))))
8	7	3anbi1d 1449	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → ((𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁) ↔ (𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁)))
9	8	rexbidv 3165	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁) ↔ ∃𝑝 ∈ ℕ0s (𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁)))
10	9	2rexbidv 3206	. . . . . 6 ⊢ (𝑧 = 𝑤 → (∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁) ↔ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁)))
11		oveq1 7367	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (𝑥 +s (𝑦 /su (2s↑s𝑝))) = (𝑎 +s (𝑦 /su (2s↑s𝑝))))
12	11	eqeq2d 2752	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ↔ 𝑤 = (𝑎 +s (𝑦 /su (2s↑s𝑝)))))
13		oveq1 7367	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (𝑥 +s 𝑝) = (𝑎 +s 𝑝))
14	13	breq1d 5085	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → ((𝑥 +s 𝑝) <s 𝑁 ↔ (𝑎 +s 𝑝) <s 𝑁))
15	12, 14	3anbi13d 1447	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → ((𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁) ↔ (𝑤 = (𝑎 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑎 +s 𝑝) <s 𝑁)))
16	15	rexbidv 3165	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (∃𝑝 ∈ ℕ0s (𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁) ↔ ∃𝑝 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑎 +s 𝑝) <s 𝑁)))
17		oveq1 7367	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑏 → (𝑦 /su (2s↑s𝑝)) = (𝑏 /su (2s↑s𝑝)))
18	17	oveq2d 7376	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑏 → (𝑎 +s (𝑦 /su (2s↑s𝑝))) = (𝑎 +s (𝑏 /su (2s↑s𝑝))))
19	18	eqeq2d 2752	. . . . . . . . . 10 ⊢ (𝑦 = 𝑏 → (𝑤 = (𝑎 +s (𝑦 /su (2s↑s𝑝))) ↔ 𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑝)))))
20		breq1 5078	. . . . . . . . . 10 ⊢ (𝑦 = 𝑏 → (𝑦 <s (2s↑s𝑝) ↔ 𝑏 <s (2s↑s𝑝)))
21	19, 20	3anbi12d 1446	. . . . . . . . 9 ⊢ (𝑦 = 𝑏 → ((𝑤 = (𝑎 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑎 +s 𝑝) <s 𝑁) ↔ (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑝))) ∧ 𝑏 <s (2s↑s𝑝) ∧ (𝑎 +s 𝑝) <s 𝑁)))
22	21	rexbidv 3165	. . . . . . . 8 ⊢ (𝑦 = 𝑏 → (∃𝑝 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑎 +s 𝑝) <s 𝑁) ↔ ∃𝑝 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑝))) ∧ 𝑏 <s (2s↑s𝑝) ∧ (𝑎 +s 𝑝) <s 𝑁)))
23		oveq2 7368	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑞 → (2s↑s𝑝) = (2s↑s𝑞))
24		oveq2 7368	. . . . . . . . . . . . 13 ⊢ ((2s↑s𝑝) = (2s↑s𝑞) → (𝑏 /su (2s↑s𝑝)) = (𝑏 /su (2s↑s𝑞)))
25	23, 24	syl 17	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑞 → (𝑏 /su (2s↑s𝑝)) = (𝑏 /su (2s↑s𝑞)))
26	25	oveq2d 7376	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑞 → (𝑎 +s (𝑏 /su (2s↑s𝑝))) = (𝑎 +s (𝑏 /su (2s↑s𝑞))))
27	26	eqeq2d 2752	. . . . . . . . . 10 ⊢ (𝑝 = 𝑞 → (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑝))) ↔ 𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞)))))
28	23	breq2d 5087	. . . . . . . . . 10 ⊢ (𝑝 = 𝑞 → (𝑏 <s (2s↑s𝑝) ↔ 𝑏 <s (2s↑s𝑞)))
29		oveq2 7368	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑞 → (𝑎 +s 𝑝) = (𝑎 +s 𝑞))
30	29	breq1d 5085	. . . . . . . . . 10 ⊢ (𝑝 = 𝑞 → ((𝑎 +s 𝑝) <s 𝑁 ↔ (𝑎 +s 𝑞) <s 𝑁))
31	27, 28, 30	3anbi123d 1445	. . . . . . . . 9 ⊢ (𝑝 = 𝑞 → ((𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑝))) ∧ 𝑏 <s (2s↑s𝑝) ∧ (𝑎 +s 𝑝) <s 𝑁) ↔ (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s 𝑁)))
32	31	cbvrexvw 3220	. . . . . . . 8 ⊢ (∃𝑝 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑝))) ∧ 𝑏 <s (2s↑s𝑝) ∧ (𝑎 +s 𝑝) <s 𝑁) ↔ ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s 𝑁))
33	22, 32	bitrdi 289	. . . . . . 7 ⊢ (𝑦 = 𝑏 → (∃𝑝 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑎 +s 𝑝) <s 𝑁) ↔ ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s 𝑁)))
34	16, 33	cbvrex2vw 3224	. . . . . 6 ⊢ (∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁) ↔ ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s 𝑁))
35	10, 34	bitrdi 289	. . . . 5 ⊢ (𝑧 = 𝑤 → (∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁) ↔ ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s 𝑁)))
36	6, 35	orbi12d 925	. . . 4 ⊢ (𝑧 = 𝑤 → ((𝑧 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁)) ↔ (𝑤 = 𝑁 ∨ ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s 𝑁))))
37	5, 36	imbi12d 346	. . 3 ⊢ (𝑧 = 𝑤 → (((( bday ‘𝑧) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁))) ↔ ((( bday ‘𝑤) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑤) → (𝑤 = 𝑁 ∨ ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s 𝑁)))))
38	37	cbvralvw 3219	. 2 ⊢ (∀𝑧 ∈ No ((( bday ‘𝑧) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁))) ↔ ∀𝑤 ∈ No ((( bday ‘𝑤) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑤) → (𝑤 = 𝑁 ∨ ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s 𝑁))))
39	1, 38	sylib 220	1 ⊢ (𝜑 → ∀𝑤 ∈ No ((( bday ‘𝑤) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑤) → (𝑤 = 𝑁 ∨ ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s 𝑁))))

Syntax hints:   → wi 4   ∧ wa 397   ∨ wo 854   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121  ∀wral 3055  ∃wrex 3065   ⊆ wss 3885   class class class wbr 5075  ‘cfv 6489  (class class class)co 7360   No csur 27625   <s clts 27626   bday cbday 27627   ≤s cles 27730   0_s c0s 27819   +_s cadds 27973   /_su cdivs 28201  ℕ_0scn0s 28326  2_sc2s 28424  ↑_scexps 28426

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-iota 6445  df-fv 6497  df-ov 7363

This theorem is referenced by:  bdayfinbndlem2  28482