Theorem raliunxp 4823

Description: Write a double restricted quantification as one universal quantifier. In this version of ralxp 4825, 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 4821	. . . . . 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 2862	. . . . . . . 8 ⊢ y ∈ V
10		vex 2862	. . . . . . . 8 ⊢ z ∈ V
11	9, 10	opex 4588	. . . . . . 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 2885	. . . . . 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 2619	. 2 ⊢ (∀x ∈ ∪ y ∈ A ({y} × B)φ ↔ ∀x(x ∈ ∪y ∈ A ({y} × B) → φ))
20		r2al 2651	. 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 2614  {csn 3737  ∪ciun 3969  ⟨cop 4561   × cxp 4770

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-csb 3137  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-iun 3971  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-xp 4784

This theorem is referenced by:  rexiunxp  4824  ralxp  4825  fmpt2x  5730