Theorem raliunxp 4824

Description: Write a double restricted quantification as one universal quantifier. In this version of ralxp 4826, B(y) is not assumed to be constant. (Contributed by Mario Carneiro, 29-Dec-2014.)

Hypothesis

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

Assertion

Ref	Expression
raliunxp	⊢ (∀x ∈ ∪ y ∈ A ({y} × B)φ ↔ ∀y ∈ A ∀z ∈ B ψ)

Distinct variable groups:   x,y,z,A   x,B,z   φ,y,z   ψ,x

Allowed substitution hints:   φ(x)   ψ(y,z)   B(y)

Proof of Theorem raliunxp

Step	Hyp	Ref	Expression
1		eliunxp 4822	. . . . . 6 ⊢ (x ∈ ∪y ∈ A ({y} × B) ↔ ∃y∃z(x = ⟨y, z⟩ ∧ (y ∈ A ∧ z ∈ B)))
2	1	imbi1i 315	. . . . 5 ⊢ ((x ∈ ∪y ∈ A ({y} × B) → φ) ↔ (∃y∃z(x = ⟨y, z⟩ ∧ (y ∈ A ∧ z ∈ B)) → φ))
3		19.23vv 1892	. . . . 5 ⊢ (∀y∀z((x = ⟨y, z⟩ ∧ (y ∈ A ∧ z ∈ B)) → φ) ↔ (∃y∃z(x = ⟨y, z⟩ ∧ (y ∈ A ∧ z ∈ B)) → φ))
4	2, 3	bitr4i 243	. . . 4 ⊢ ((x ∈ ∪y ∈ A ({y} × B) → φ) ↔ ∀y∀z((x = ⟨y, z⟩ ∧ (y ∈ A ∧ z ∈ B)) → φ))
5	4	albii 1566	. . 3 ⊢ (∀x(x ∈ ∪y ∈ A ({y} × B) → φ) ↔ ∀x∀y∀z((x = ⟨y, z⟩ ∧ (y ∈ A ∧ z ∈ B)) → φ))
6		alrot3 1738	. . . 4 ⊢ (∀x∀y∀z((x = ⟨y, z⟩ ∧ (y ∈ A ∧ z ∈ B)) → φ) ↔ ∀y∀z∀x((x = ⟨y, z⟩ ∧ (y ∈ A ∧ z ∈ B)) → φ))
7		impexp 433	. . . . . . 7 ⊢ (((x = ⟨y, z⟩ ∧ (y ∈ A ∧ z ∈ B)) → φ) ↔ (x = ⟨y, z⟩ → ((y ∈ A ∧ z ∈ B) → φ)))
8	7	albii 1566	. . . . . 6 ⊢ (∀x((x = ⟨y, z⟩ ∧ (y ∈ A ∧ z ∈ B)) → φ) ↔ ∀x(x = ⟨y, z⟩ → ((y ∈ A ∧ z ∈ B) → φ)))
9		vex 2863	. . . . . . . 8 ⊢ y ∈ V
10		vex 2863	. . . . . . . 8 ⊢ z ∈ V
11	9, 10	opex 4589	. . . . . . 7 ⊢ ⟨y, z⟩ ∈ V
12		raliunxp.1	. . . . . . . 8 ⊢ (x = ⟨y, z⟩ → (φ ↔ ψ))
13	12	imbi2d 307	. . . . . . 7 ⊢ (x = ⟨y, z⟩ → (((y ∈ A ∧ z ∈ B) → φ) ↔ ((y ∈ A ∧ z ∈ B) → ψ)))
14	11, 13	ceqsalv 2886	. . . . . 6 ⊢ (∀x(x = ⟨y, z⟩ → ((y ∈ A ∧ z ∈ B) → φ)) ↔ ((y ∈ A ∧ z ∈ B) → ψ))
15	8, 14	bitri 240	. . . . 5 ⊢ (∀x((x = ⟨y, z⟩ ∧ (y ∈ A ∧ z ∈ B)) → φ) ↔ ((y ∈ A ∧ z ∈ B) → ψ))
16	15	2albii 1567	. . . 4 ⊢ (∀y∀z∀x((x = ⟨y, z⟩ ∧ (y ∈ A ∧ z ∈ B)) → φ) ↔ ∀y∀z((y ∈ A ∧ z ∈ B) → ψ))
17	6, 16	bitri 240	. . 3 ⊢ (∀x∀y∀z((x = ⟨y, z⟩ ∧ (y ∈ A ∧ z ∈ B)) → φ) ↔ ∀y∀z((y ∈ A ∧ z ∈ B) → ψ))
18	5, 17	bitri 240	. 2 ⊢ (∀x(x ∈ ∪y ∈ A ({y} × B) → φ) ↔ ∀y∀z((y ∈ A ∧ z ∈ B) → ψ))
19		df-ral 2620	. 2 ⊢ (∀x ∈ ∪ y ∈ A ({y} × B)φ ↔ ∀x(x ∈ ∪y ∈ A ({y} × B) → φ))
20		r2al 2652	. 2 ⊢ (∀y ∈ A ∀z ∈ B ψ ↔ ∀y∀z((y ∈ A ∧ z ∈ B) → ψ))
21	18, 19, 20	3bitr4i 268	1 ⊢ (∀x ∈ ∪ y ∈ A ({y} × B)φ ↔ ∀y ∈ A ∀z ∈ B ψ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∀wral 2615  {csn 3738  ∪ciun 3970  ⟨cop 4562   × cxp 4771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-csb 3138  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-iun 3972  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-xp 4785

This theorem is referenced by:  rexiunxp  4825  ralxp  4826  fmpt2x  5731