Theorem ralxp 2456

Description: Universal quantification restricted to a cross product is equivalent to a double restricted quantification. The hypothesis specifies an implicit substitution.

Hypothesis

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

Assertion

Ref	Expression
ralxp	⊢ (∀x ∈ (A × B)φ ↔ ∀y ∈ A ∀z ∈ B ψ)

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

Proof of Theorem ralxp

Step	Hyp	Ref	Expression
1		ralxp.1	. . . . . . 7 ⊢ (x = ⟨y, z⟩ → (φ ↔ ψ))
2	1	rcla4v 1402	. . . . . 6 ⊢ (∀x ∈ (A × B)φ → (⟨y, z⟩ ∈ (A × B) → ψ))
3		visset 1350	. . . . . . 7 ⊢ z ∈ V
4	3	opelxp 2452	. . . . . 6 ⊢ (⟨y, z⟩ ∈ (A × B) ↔ (y ∈ A ∧ z ∈ B))
5	2, 4	syl5ibr 182	. . . . 5 ⊢ (∀x ∈ (A × B)φ → ((y ∈ A ∧ z ∈ B) → ψ))
6	5	exp3a 292	. . . 4 ⊢ (∀x ∈ (A × B)φ → (y ∈ A → (z ∈ B → ψ)))
7	6	r19.21adv 1262	. . 3 ⊢ (∀x ∈ (A × B)φ → (y ∈ A → ∀z ∈ B ψ))
8	7	r19.21aiv 1259	. 2 ⊢ (∀x ∈ (A × B)φ → ∀y ∈ A ∀z ∈ B ψ)
9		elxp 2442	. . . . . 6 ⊢ (x ∈ (A × B) ↔ ∃y∃z(x = ⟨y, z⟩ ∧ (y ∈ A ∧ z ∈ B)))
10		pm3.26 256	. . . . . . . 8 ⊢ ((x = ⟨y, z⟩ ∧ (y ∈ A ∧ z ∈ B)) → x = ⟨y, z⟩)
11	10	19.22i 723	. . . . . . 7 ⊢ (∃z(x = ⟨y, z⟩ ∧ (y ∈ A ∧ z ∈ B)) → ∃z x = ⟨y, z⟩)
12	11	19.22i 723	. . . . . 6 ⊢ (∃y∃z(x = ⟨y, z⟩ ∧ (y ∈ A ∧ z ∈ B)) → ∃y∃z x = ⟨y, z⟩)
13	9, 12	sylbi 174	. . . . 5 ⊢ (x ∈ (A × B) → ∃y∃z x = ⟨y, z⟩)
14		hbra1 1237	. . . . . . 7 ⊢ (∀y ∈ A ∀z ∈ B ψ → ∀y∀y ∈ A ∀z ∈ B ψ)
15		ax-17 925	. . . . . . 7 ⊢ ((x ∈ (A × B) → φ) → ∀y(x ∈ (A × B) → φ))
16	14, 15	hbim 702	. . . . . 6 ⊢ ((∀y ∈ A ∀z ∈ B ψ → (x ∈ (A × B) → φ)) → ∀y(∀y ∈ A ∀z ∈ B ψ → (x ∈ (A × B) → φ)))
17		ax-17 925	. . . . . . . . 9 ⊢ (y ∈ A → ∀z y ∈ A)
18		hbra1 1237	. . . . . . . . 9 ⊢ (∀z ∈ B ψ → ∀z∀z ∈ B ψ)
19	17, 18	hbral 1236	. . . . . . . 8 ⊢ (∀y ∈ A ∀z ∈ B ψ → ∀z∀y ∈ A ∀z ∈ B ψ)
20		ax-17 925	. . . . . . . 8 ⊢ ((x ∈ (A × B) → φ) → ∀z(x ∈ (A × B) → φ))
21	19, 20	hbim 702	. . . . . . 7 ⊢ ((∀y ∈ A ∀z ∈ B ψ → (x ∈ (A × B) → φ)) → ∀z(∀y ∈ A ∀z ∈ B ψ → (x ∈ (A × B) → φ)))
22		eleq1 1149	. . . . . . . . . 10 ⊢ (x = ⟨y, z⟩ → (x ∈ (A × B) ↔ ⟨y, z⟩ ∈ (A × B)))
23	22, 4	syl6bb 414	. . . . . . . . 9 ⊢ (x = ⟨y, z⟩ → (x ∈ (A × B) ↔ (y ∈ A ∧ z ∈ B)))
24	23, 1	imbi12d 474	. . . . . . . 8 ⊢ (x = ⟨y, z⟩ → ((x ∈ (A × B) → φ) ↔ ((y ∈ A ∧ z ∈ B) → ψ)))
25		ra42 1245	. . . . . . . 8 ⊢ (∀y ∈ A ∀z ∈ B ψ → ((y ∈ A ∧ z ∈ B) → ψ))
26	24, 25	syl5bir 184	. . . . . . 7 ⊢ (x = ⟨y, z⟩ → (∀y ∈ A ∀z ∈ B ψ → (x ∈ (A × B) → φ)))
27	21, 26	19.23ai 746	. . . . . 6 ⊢ (∃z x = ⟨y, z⟩ → (∀y ∈ A ∀z ∈ B ψ → (x ∈ (A × B) → φ)))
28	16, 27	19.23ai 746	. . . . 5 ⊢ (∃y∃z x = ⟨y, z⟩ → (∀y ∈ A ∀z ∈ B ψ → (x ∈ (A × B) → φ)))
29	13, 28	syl 12	. . . 4 ⊢ (x ∈ (A × B) → (∀y ∈ A ∀z ∈ B ψ → (x ∈ (A × B) → φ)))
30	29	pm2.43b 61	. . 3 ⊢ (∀y ∈ A ∀z ∈ B ψ → (x ∈ (A × B) → φ))
31	30	r19.21aiv 1259	. 2 ⊢ (∀y ∈ A ∀z ∈ B ψ → ∀x ∈ (A × B)φ)
32	8, 31	impbi 139	1 ⊢ (∀x ∈ (A × B)φ ↔ ∀y ∈ A ∀z ∈ B ψ)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ⟨cop 1810   × cxp 2408

This theorem is referenced by:  ffnoprval 3041  f1stres 3096  df1st2 3098

This theorem was proved 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-ral 1205  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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