Theorem cbvop 2473

Description: Change restricted bound variable to two restricted bound variables.

Hypothesis

Ref	Expression
cbvop.1	⊢ (x = ⟨y, z⟩ → (φ ↔ ψ))

Assertion

Ref	Expression
cbvop	⊢ (∃x ∈ (A × B)φ ↔ ∃y ∈ A ∃z ∈ B ψ)

Distinct variable group(s):   φ,y,z   ψ,x   x,y,z,A   x,B,y,z

Proof of Theorem cbvop

Step	Hyp	Ref	Expression
1		anass 336	. . . . 5 ⊢ (((∃y∃z x = ⟨y, z⟩ ∧ x ∈ (A × B)) ∧ φ) ↔ (∃y∃z x = ⟨y, z⟩ ∧ (x ∈ (A × B) ∧ φ)))
2		elxp3 2460	. . . . . . 7 ⊢ (x ∈ (A × B) ↔ ∃y∃z(⟨y, z⟩ = x ∧ ⟨y, z⟩ ∈ (A × B)))
3		eleq1 1149	. . . . . . . . . 10 ⊢ (⟨y, z⟩ = x → (⟨y, z⟩ ∈ (A × B) ↔ x ∈ (A × B)))
4	3	pm5.32i 489	. . . . . . . . 9 ⊢ ((⟨y, z⟩ = x ∧ ⟨y, z⟩ ∈ (A × B)) ↔ (⟨y, z⟩ = x ∧ x ∈ (A × B)))
5		cleqcom 1103	. . . . . . . . . 10 ⊢ (x = ⟨y, z⟩ ↔ ⟨y, z⟩ = x)
6	5	anbi1i 368	. . . . . . . . 9 ⊢ ((x = ⟨y, z⟩ ∧ x ∈ (A × B)) ↔ (⟨y, z⟩ = x ∧ x ∈ (A × B)))
7	4, 6	bitr4 154	. . . . . . . 8 ⊢ ((⟨y, z⟩ = x ∧ ⟨y, z⟩ ∈ (A × B)) ↔ (x = ⟨y, z⟩ ∧ x ∈ (A × B)))
8	7	bi2ex 734	. . . . . . 7 ⊢ (∃y∃z(⟨y, z⟩ = x ∧ ⟨y, z⟩ ∈ (A × B)) ↔ ∃y∃z(x = ⟨y, z⟩ ∧ x ∈ (A × B)))
9		19.41vv 964	. . . . . . 7 ⊢ (∃y∃z(x = ⟨y, z⟩ ∧ x ∈ (A × B)) ↔ (∃y∃z x = ⟨y, z⟩ ∧ x ∈ (A × B)))
10	2, 8, 9	3bitr 155	. . . . . 6 ⊢ (x ∈ (A × B) ↔ (∃y∃z x = ⟨y, z⟩ ∧ x ∈ (A × B)))
11	10	anbi1i 368	. . . . 5 ⊢ ((x ∈ (A × B) ∧ φ) ↔ ((∃y∃z x = ⟨y, z⟩ ∧ x ∈ (A × B)) ∧ φ))
12		19.41vv 964	. . . . 5 ⊢ (∃y∃z(x = ⟨y, z⟩ ∧ (x ∈ (A × B) ∧ φ)) ↔ (∃y∃z x = ⟨y, z⟩ ∧ (x ∈ (A × B) ∧ φ)))
13	1, 11, 12	3bitr4 158	. . . 4 ⊢ ((x ∈ (A × B) ∧ φ) ↔ ∃y∃z(x = ⟨y, z⟩ ∧ (x ∈ (A × B) ∧ φ)))
14	13	biex 733	. . 3 ⊢ (∃x(x ∈ (A × B) ∧ φ) ↔ ∃x∃y∃z(x = ⟨y, z⟩ ∧ (x ∈ (A × B) ∧ φ)))
15		exrot3 777	. . 3 ⊢ (∃x∃y∃z(x = ⟨y, z⟩ ∧ (x ∈ (A × B) ∧ φ)) ↔ ∃y∃z∃x(x = ⟨y, z⟩ ∧ (x ∈ (A × B) ∧ φ)))
16		opex 1893	. . . . 5 ⊢ ⟨y, z⟩ ∈ V
17		eleq1 1149	. . . . . 6 ⊢ (x = ⟨y, z⟩ → (x ∈ (A × B) ↔ ⟨y, z⟩ ∈ (A × B)))
18		cbvop.1	. . . . . 6 ⊢ (x = ⟨y, z⟩ → (φ ↔ ψ))
19	17, 18	anbi12d 476	. . . . 5 ⊢ (x = ⟨y, z⟩ → ((x ∈ (A × B) ∧ φ) ↔ (⟨y, z⟩ ∈ (A × B) ∧ ψ)))
20	16, 19	ceqsexv 1371	. . . 4 ⊢ (∃x(x = ⟨y, z⟩ ∧ (x ∈ (A × B) ∧ φ)) ↔ (⟨y, z⟩ ∈ (A × B) ∧ ψ))
21	20	bi2ex 734	. . 3 ⊢ (∃y∃z∃x(x = ⟨y, z⟩ ∧ (x ∈ (A × B) ∧ φ)) ↔ ∃y∃z(⟨y, z⟩ ∈ (A × B) ∧ ψ))
22	14, 15, 21	3bitr 155	. 2 ⊢ (∃x(x ∈ (A × B) ∧ φ) ↔ ∃y∃z(⟨y, z⟩ ∈ (A × B) ∧ ψ))
23		df-rex 1206	. 2 ⊢ (∃x ∈ (A × B)φ ↔ ∃x(x ∈ (A × B) ∧ φ))
24		r2ex 1241	. . 3 ⊢ (∃y ∈ A ∃z ∈ B ψ ↔ ∃y∃z((y ∈ A ∧ z ∈ B) ∧ ψ))
25		visset 1350	. . . . . 6 ⊢ z ∈ V
26	25	opelxp 2452	. . . . 5 ⊢ (⟨y, z⟩ ∈ (A × B) ↔ (y ∈ A ∧ z ∈ B))
27	26	anbi1i 368	. . . 4 ⊢ ((⟨y, z⟩ ∈ (A × B) ∧ ψ) ↔ ((y ∈ A ∧ z ∈ B) ∧ ψ))
28	27	bi2ex 734	. . 3 ⊢ (∃y∃z(⟨y, z⟩ ∈ (A × B) ∧ ψ) ↔ ∃y∃z((y ∈ A ∧ z ∈ B) ∧ ψ))
29	24, 28	bitr4 154	. 2 ⊢ (∃y ∈ A ∃z ∈ B ψ ↔ ∃y∃z(⟨y, z⟩ ∈ (A × B) ∧ ψ))
30	22, 23, 29	3bitr4 158	1 ⊢ (∃x ∈ (A × B)φ ↔ ∃y ∈ A ∃z ∈ B ψ)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  ⟨cop 1810   × cxp 2408

This theorem is referenced by:  elrnoprab 3054  oprvalex 3055

This theorem was provFd from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098  df-xp 2424

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