Theorem ralxpf 4828

Description: Version of ralxp 4826 with bound-variable hypotheses. (Contributed by NM, 18-Aug-2006.) (Revised by set.mm contributors, 20-Dec-2008.)

Hypotheses

Ref	Expression
ralxpf.1	⊢ Ⅎyφ
ralxpf.2	⊢ Ⅎzφ
ralxpf.3	⊢ Ⅎxψ
ralxpf.4	⊢ (x = ⟨y, z⟩ → (φ ↔ ψ))

Assertion

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

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

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

Proof of Theorem ralxpf

Dummy variables v u w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvralsv 2847	. 2 ⊢ (∀x ∈ (A × B)φ ↔ ∀w ∈ (A × B)[w / x]φ)
2		cbvralsv 2847	. . . 4 ⊢ (∀z ∈ B [u / y]ψ ↔ ∀v ∈ B [v / z][u / y]ψ)
3	2	ralbii 2639	. . 3 ⊢ (∀u ∈ A ∀z ∈ B [u / y]ψ ↔ ∀u ∈ A ∀v ∈ B [v / z][u / y]ψ)
4		nfv 1619	. . . 4 ⊢ Ⅎu∀z ∈ B ψ
5		nfcv 2490	. . . . 5 ⊢ ℲyB
6		nfv 1619	. . . . . 6 ⊢ Ⅎuψ
7	6	nfs1 2044	. . . . 5 ⊢ Ⅎy[u / y]ψ
8	5, 7	nfral 2668	. . . 4 ⊢ Ⅎy∀z ∈ B [u / y]ψ
9		sbequ12 1919	. . . . 5 ⊢ (y = u → (ψ ↔ [u / y]ψ))
10	9	ralbidv 2635	. . . 4 ⊢ (y = u → (∀z ∈ B ψ ↔ ∀z ∈ B [u / y]ψ))
11	4, 8, 10	cbvral 2832	. . 3 ⊢ (∀y ∈ A ∀z ∈ B ψ ↔ ∀u ∈ A ∀z ∈ B [u / y]ψ)
12		vex 2863	. . . . . 6 ⊢ u ∈ V
13		vex 2863	. . . . . 6 ⊢ v ∈ V
14	12, 13	eqvinop 4607	. . . . 5 ⊢ (w = ⟨u, v⟩ ↔ ∃y∃z(w = ⟨y, z⟩ ∧ ⟨y, z⟩ = ⟨u, v⟩))
15		ralxpf.1	. . . . . . . 8 ⊢ Ⅎyφ
16	15	nfsb 2109	. . . . . . 7 ⊢ Ⅎy[w / x]φ
17	7	nfsb 2109	. . . . . . 7 ⊢ Ⅎy[v / z][u / y]ψ
18	16, 17	nfbi 1834	. . . . . 6 ⊢ Ⅎy([w / x]φ ↔ [v / z][u / y]ψ)
19		ralxpf.2	. . . . . . . . 9 ⊢ Ⅎzφ
20	19	nfsb 2109	. . . . . . . 8 ⊢ Ⅎz[w / x]φ
21		nfv 1619	. . . . . . . . 9 ⊢ Ⅎv[u / y]ψ
22	21	nfs1 2044	. . . . . . . 8 ⊢ Ⅎz[v / z][u / y]ψ
23	20, 22	nfbi 1834	. . . . . . 7 ⊢ Ⅎz([w / x]φ ↔ [v / z][u / y]ψ)
24		ralxpf.3	. . . . . . . . 9 ⊢ Ⅎxψ
25		ralxpf.4	. . . . . . . . 9 ⊢ (x = ⟨y, z⟩ → (φ ↔ ψ))
26	24, 25	sbhypf 2905	. . . . . . . 8 ⊢ (w = ⟨y, z⟩ → ([w / x]φ ↔ ψ))
27		opth 4603	. . . . . . . . 9 ⊢ (⟨y, z⟩ = ⟨u, v⟩ ↔ (y = u ∧ z = v))
28		sbequ12 1919	. . . . . . . . . 10 ⊢ (z = v → ([u / y]ψ ↔ [v / z][u / y]ψ))
29	9, 28	sylan9bb 680	. . . . . . . . 9 ⊢ ((y = u ∧ z = v) → (ψ ↔ [v / z][u / y]ψ))
30	27, 29	sylbi 187	. . . . . . . 8 ⊢ (⟨y, z⟩ = ⟨u, v⟩ → (ψ ↔ [v / z][u / y]ψ))
31	26, 30	sylan9bb 680	. . . . . . 7 ⊢ ((w = ⟨y, z⟩ ∧ ⟨y, z⟩ = ⟨u, v⟩) → ([w / x]φ ↔ [v / z][u / y]ψ))
32	23, 31	exlimi 1803	. . . . . 6 ⊢ (∃z(w = ⟨y, z⟩ ∧ ⟨y, z⟩ = ⟨u, v⟩) → ([w / x]φ ↔ [v / z][u / y]ψ))
33	18, 32	exlimi 1803	. . . . 5 ⊢ (∃y∃z(w = ⟨y, z⟩ ∧ ⟨y, z⟩ = ⟨u, v⟩) → ([w / x]φ ↔ [v / z][u / y]ψ))
34	14, 33	sylbi 187	. . . 4 ⊢ (w = ⟨u, v⟩ → ([w / x]φ ↔ [v / z][u / y]ψ))
35	34	ralxp 4826	. . 3 ⊢ (∀w ∈ (A × B)[w / x]φ ↔ ∀u ∈ A ∀v ∈ B [v / z][u / y]ψ)
36	3, 11, 35	3bitr4ri 269	. 2 ⊢ (∀w ∈ (A × B)[w / x]φ ↔ ∀y ∈ A ∀z ∈ B ψ)
37	1, 36	bitri 240	1 ⊢ (∀x ∈ (A × B)φ ↔ ∀y ∈ A ∀z ∈ B ψ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544   = wceq 1642  [wsb 1648  ∀wral 2615  ⟨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:  rexxpf  4829