Theorem cdleme40v 41243

Description: Part of proof of Lemma E in [Crawley] p. 113. Change bound variables in ⦋𝑆 / 𝑢⦌𝑉 (but we use ⦋𝑅 / 𝑢⦌𝑉 for convenience since we have its hypotheses available). (Contributed by NM, 18-Mar-2013.)

Hypotheses

Ref	Expression
cdleme40.b	⊢ 𝐵 = (Base‘𝐾)
cdleme40.l	⊢ ≤ = (le‘𝐾)
cdleme40.j	⊢ ∨ = (join‘𝐾)
cdleme40.m	⊢ ∧ = (meet‘𝐾)
cdleme40.a	⊢ 𝐴 = (Atoms‘𝐾)
cdleme40.h	⊢ 𝐻 = (LHyp‘𝐾)
cdleme40.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdleme40.e	⊢ 𝐸 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
cdleme40.g	⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
cdleme40.i	⊢ 𝐼 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐺))
cdleme40.n	⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷)
cdleme40.d	⊢ 𝐷 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)))
cdleme40r.y	⊢ 𝑌 = ((𝑢 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑢) ∧ 𝑊)))
cdleme40r.t	⊢ 𝑇 = ((𝑣 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑣) ∧ 𝑊)))
cdleme40r.x	⊢ 𝑋 = ((𝑃 ∨ 𝑄) ∧ (𝑇 ∨ ((𝑢 ∨ 𝑣) ∧ 𝑊)))
cdleme40r.o	⊢ 𝑂 = (℩𝑧 ∈ 𝐵 ∀𝑣 ∈ 𝐴 ((¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)) → 𝑧 = 𝑋))
cdleme40r.v	⊢ 𝑉 = if(𝑢 ≤ (𝑃 ∨ 𝑄), 𝑂, 𝑌)

Assertion

Ref	Expression
cdleme40v	⊢ (𝑅 ∈ 𝐴 → ⦋𝑅 / 𝑠⦌𝑁 = ⦋𝑅 / 𝑢⦌𝑉)

Distinct variable groups:   ∧ ,𝑠,𝑡,𝑦   𝑧,𝑈   𝑧,𝑅   𝑇,𝑠,𝑡,𝑦   𝑅,𝑠,𝑡,𝑣,𝑦   𝑄,𝑠,𝑡,𝑦   𝑧,𝐾   𝑢,𝑃,𝑧   𝑢,𝑄,𝑣,𝑧   𝑧,𝐻   𝑃,𝑠,𝑡,𝑣,𝑦   𝐸,𝑠   𝑢,𝑊,𝑧,𝑠,𝑡,𝑣,𝑦   𝐵,𝑠,𝑡,𝑦,𝑢,𝑣,𝑧   𝑦,𝑌   𝑢, ∨ ,𝑧,𝑠,𝑡,𝑣,𝑦   𝑢, ≤ ,𝑧,𝑠,𝑡,𝑣,𝑦   𝐴,𝑠,𝑡,𝑣,𝑦   𝑢, ∧ ,𝑣,𝑧   𝑡,𝑈,𝑣,𝑦   𝑡,𝐾,𝑣,𝑦   𝑡,𝐻,𝑣,𝑦   𝑢,𝐴,𝑧   𝑢,𝑇   𝑣,𝐸,𝑧   𝑢,𝑁,𝑣   𝑢,𝑅   𝑉,𝑠   𝑡,𝑋,𝑦   𝑢,𝑠,𝑧,𝑡,𝑦

Allowed substitution hints:   𝐷(𝑦,𝑧,𝑣,𝑢,𝑡,𝑠)   𝑇(𝑧,𝑣)   𝑈(𝑢,𝑠)   𝐸(𝑦,𝑢,𝑡)   𝐺(𝑦,𝑧,𝑣,𝑢,𝑡,𝑠)   𝐻(𝑢,𝑠)   𝐼(𝑦,𝑧,𝑣,𝑢,𝑡,𝑠)   𝐾(𝑢,𝑠)   𝑁(𝑦,𝑧,𝑡,𝑠)   𝑂(𝑦,𝑧,𝑣,𝑢,𝑡,𝑠)   𝑉(𝑦,𝑧,𝑣,𝑢,𝑡)   𝑋(𝑧,𝑣,𝑢,𝑠)   𝑌(𝑧,𝑣,𝑢,𝑡,𝑠)

Proof of Theorem cdleme40v

Step	Hyp	Ref	Expression
1		breq1 5112	. . . . 5 ⊢ (𝑠 = 𝑢 → (𝑠 ≤ (𝑃 ∨ 𝑄) ↔ 𝑢 ≤ (𝑃 ∨ 𝑄)))
2		cdleme40.g	. . . . . . . . . . . 12 ⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
3		oveq1 7417	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑢 → (𝑠 ∨ 𝑡) = (𝑢 ∨ 𝑡))
4	3	oveq1d 7425	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑢 → ((𝑠 ∨ 𝑡) ∧ 𝑊) = ((𝑢 ∨ 𝑡) ∧ 𝑊))
5	4	oveq2d 7426	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑢 → (𝐸 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)) = (𝐸 ∨ ((𝑢 ∨ 𝑡) ∧ 𝑊)))
6	5	oveq2d 7426	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑢 → ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊))) = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑢 ∨ 𝑡) ∧ 𝑊))))
7	2, 6	eqtrid 2810	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑢 → 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑢 ∨ 𝑡) ∧ 𝑊))))
8	7	eqeq2d 2774	. . . . . . . . . 10 ⊢ (𝑠 = 𝑢 → (𝑦 = 𝐺 ↔ 𝑦 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑢 ∨ 𝑡) ∧ 𝑊)))))
9	8	imbi2d 343	. . . . . . . . 9 ⊢ (𝑠 = 𝑢 → (((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐺) ↔ ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑢 ∨ 𝑡) ∧ 𝑊))))))
10	9	ralbidv 3188	. . . . . . . 8 ⊢ (𝑠 = 𝑢 → (∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐺) ↔ ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑢 ∨ 𝑡) ∧ 𝑊))))))
11	10	riotabidv 7369	. . . . . . 7 ⊢ (𝑠 = 𝑢 → (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐺)) = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑢 ∨ 𝑡) ∧ 𝑊))))))
12		eqeq1 2767	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝑦 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑢 ∨ 𝑡) ∧ 𝑊))) ↔ 𝑧 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑢 ∨ 𝑡) ∧ 𝑊)))))
13	12	imbi2d 343	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑢 ∨ 𝑡) ∧ 𝑊)))) ↔ ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑧 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑢 ∨ 𝑡) ∧ 𝑊))))))
14	13	ralbidv 3188	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑢 ∨ 𝑡) ∧ 𝑊)))) ↔ ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑧 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑢 ∨ 𝑡) ∧ 𝑊))))))
15		breq1 5112	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑣 → (𝑡 ≤ 𝑊 ↔ 𝑣 ≤ 𝑊))
16	15	notbid 321	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑣 → (¬ 𝑡 ≤ 𝑊 ↔ ¬ 𝑣 ≤ 𝑊))
17		breq1 5112	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑣 → (𝑡 ≤ (𝑃 ∨ 𝑄) ↔ 𝑣 ≤ (𝑃 ∨ 𝑄)))
18	17	notbid 321	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑣 → (¬ 𝑡 ≤ (𝑃 ∨ 𝑄) ↔ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))
19	16, 18	anbi12d 643	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑣 → ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ↔ (¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄))))
20		oveq1 7417	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑣 → (𝑡 ∨ 𝑈) = (𝑣 ∨ 𝑈))
21		oveq2 7418	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑣 → (𝑃 ∨ 𝑡) = (𝑃 ∨ 𝑣))
22	21	oveq1d 7425	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑣 → ((𝑃 ∨ 𝑡) ∧ 𝑊) = ((𝑃 ∨ 𝑣) ∧ 𝑊))
23	22	oveq2d 7426	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑣 → (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)) = (𝑄 ∨ ((𝑃 ∨ 𝑣) ∧ 𝑊)))
24	20, 23	oveq12d 7428	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑣 → ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) = ((𝑣 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑣) ∧ 𝑊))))
25		cdleme40.e	. . . . . . . . . . . . . . . 16 ⊢ 𝐸 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
26		cdleme40r.t	. . . . . . . . . . . . . . . 16 ⊢ 𝑇 = ((𝑣 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑣) ∧ 𝑊)))
27	24, 25, 26	3eqtr4g 2823	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑣 → 𝐸 = 𝑇)
28		oveq2 7418	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑣 → (𝑢 ∨ 𝑡) = (𝑢 ∨ 𝑣))
29	28	oveq1d 7425	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑣 → ((𝑢 ∨ 𝑡) ∧ 𝑊) = ((𝑢 ∨ 𝑣) ∧ 𝑊))
30	27, 29	oveq12d 7428	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑣 → (𝐸 ∨ ((𝑢 ∨ 𝑡) ∧ 𝑊)) = (𝑇 ∨ ((𝑢 ∨ 𝑣) ∧ 𝑊)))
31	30	oveq2d 7426	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑣 → ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑢 ∨ 𝑡) ∧ 𝑊))) = ((𝑃 ∨ 𝑄) ∧ (𝑇 ∨ ((𝑢 ∨ 𝑣) ∧ 𝑊))))
32		cdleme40r.x	. . . . . . . . . . . . 13 ⊢ 𝑋 = ((𝑃 ∨ 𝑄) ∧ (𝑇 ∨ ((𝑢 ∨ 𝑣) ∧ 𝑊)))
33	31, 32	eqtr4di 2816	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑣 → ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑢 ∨ 𝑡) ∧ 𝑊))) = 𝑋)
34	33	eqeq2d 2774	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑣 → (𝑧 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑢 ∨ 𝑡) ∧ 𝑊))) ↔ 𝑧 = 𝑋))
35	19, 34	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑡 = 𝑣 → (((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑧 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑢 ∨ 𝑡) ∧ 𝑊)))) ↔ ((¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)) → 𝑧 = 𝑋)))
36	35	cbvralvw 3243	. . . . . . . . 9 ⊢ (∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑧 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑢 ∨ 𝑡) ∧ 𝑊)))) ↔ ∀𝑣 ∈ 𝐴 ((¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)) → 𝑧 = 𝑋))
37	14, 36	bitrdi 290	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑢 ∨ 𝑡) ∧ 𝑊)))) ↔ ∀𝑣 ∈ 𝐴 ((¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)) → 𝑧 = 𝑋)))
38	37	cbvriotavw 7377	. . . . . . 7 ⊢ (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑢 ∨ 𝑡) ∧ 𝑊))))) = (℩𝑧 ∈ 𝐵 ∀𝑣 ∈ 𝐴 ((¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)) → 𝑧 = 𝑋))
39	11, 38	eqtrdi 2814	. . . . . 6 ⊢ (𝑠 = 𝑢 → (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐺)) = (℩𝑧 ∈ 𝐵 ∀𝑣 ∈ 𝐴 ((¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)) → 𝑧 = 𝑋)))
40		cdleme40.i	. . . . . 6 ⊢ 𝐼 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐺))
41		cdleme40r.o	. . . . . 6 ⊢ 𝑂 = (℩𝑧 ∈ 𝐵 ∀𝑣 ∈ 𝐴 ((¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)) → 𝑧 = 𝑋))
42	39, 40, 41	3eqtr4g 2823	. . . . 5 ⊢ (𝑠 = 𝑢 → 𝐼 = 𝑂)
43		oveq1 7417	. . . . . . 7 ⊢ (𝑠 = 𝑢 → (𝑠 ∨ 𝑈) = (𝑢 ∨ 𝑈))
44		oveq2 7418	. . . . . . . . 9 ⊢ (𝑠 = 𝑢 → (𝑃 ∨ 𝑠) = (𝑃 ∨ 𝑢))
45	44	oveq1d 7425	. . . . . . . 8 ⊢ (𝑠 = 𝑢 → ((𝑃 ∨ 𝑠) ∧ 𝑊) = ((𝑃 ∨ 𝑢) ∧ 𝑊))
46	45	oveq2d 7426	. . . . . . 7 ⊢ (𝑠 = 𝑢 → (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)) = (𝑄 ∨ ((𝑃 ∨ 𝑢) ∧ 𝑊)))
47	43, 46	oveq12d 7428	. . . . . 6 ⊢ (𝑠 = 𝑢 → ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) = ((𝑢 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑢) ∧ 𝑊))))
48		cdleme40.d	. . . . . 6 ⊢ 𝐷 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)))
49		cdleme40r.y	. . . . . 6 ⊢ 𝑌 = ((𝑢 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑢) ∧ 𝑊)))
50	47, 48, 49	3eqtr4g 2823	. . . . 5 ⊢ (𝑠 = 𝑢 → 𝐷 = 𝑌)
51	1, 42, 50	ifbieq12d 4516	. . . 4 ⊢ (𝑠 = 𝑢 → if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷) = if(𝑢 ≤ (𝑃 ∨ 𝑄), 𝑂, 𝑌))
52		cdleme40.n	. . . 4 ⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷)
53		cdleme40r.v	. . . 4 ⊢ 𝑉 = if(𝑢 ≤ (𝑃 ∨ 𝑄), 𝑂, 𝑌)
54	51, 52, 53	3eqtr4g 2823	. . 3 ⊢ (𝑠 = 𝑢 → 𝑁 = 𝑉)
55	54	cbvcsbv 3865	. 2 ⊢ ⦋𝑅 / 𝑠⦌𝑁 = ⦋𝑅 / 𝑢⦌𝑉
56	55	a1i 11	1 ⊢ (𝑅 ∈ 𝐴 → ⦋𝑅 / 𝑠⦌𝑁 = ⦋𝑅 / 𝑢⦌𝑉)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ⦋csb 3853  ifcif 4487   class class class wbr 5109  ‘cfv 6536  ℩crio 7366  (class class class)co 7410  Basecbs 17264  lecple 17312  joincjn 18362  meetcmee 18363  Atomscatm 40037  LHypclh 40758

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  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-iota 6492  df-fv 6544  df-riota 7367  df-ov 7413

This theorem is referenced by:  cdleme40w  41244