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Mirrors > Home > NFE Home > Th. List > raliunxp | GIF version |
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.) |
Ref | Expression |
---|---|
raliunxp.1 | ⊢ (x = 〈y, z〉 → (φ ↔ ψ)) |
Ref | Expression |
---|---|
raliunxp | ⊢ (∀x ∈ ∪ y ∈ A ({y} × B)φ ↔ ∀y ∈ A ∀z ∈ B ψ) |
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) |
Colors of variables: wff setvar class |
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-1 5 ax-2 6 ax-3 7 ax-mp 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 |
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