![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ralxp | Structured version Visualization version GIF version |
Description: Universal quantification restricted to a Cartesian product is equivalent to a double restricted quantification. The hypothesis specifies an implicit substitution. (Contributed by NM, 7-Feb-2004.) (Revised by Mario Carneiro, 29-Dec-2014.) |
Ref | Expression |
---|---|
ralxp.1 | ⊢ (𝑥 = 〈𝑦, 𝑧〉 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
ralxp | ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunxpconst 5760 | . . 3 ⊢ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) = (𝐴 × 𝐵) | |
2 | 1 | raleqi 3321 | . 2 ⊢ (∀𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∀𝑥 ∈ (𝐴 × 𝐵)𝜑) |
3 | ralxp.1 | . . 3 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → (𝜑 ↔ 𝜓)) | |
4 | 3 | raliunxp 5852 | . 2 ⊢ (∀𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜓) |
5 | 2, 4 | bitr3i 277 | 1 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1536 ∀wral 3058 {csn 4630 〈cop 4636 ∪ ciun 4995 × cxp 5686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-iun 4997 df-opab 5210 df-xp 5694 df-rel 5695 |
This theorem is referenced by: ralxpf 5859 reu3op 6313 f1opr 7488 ffnov 7558 eqfnov 7561 funimassov 7609 f1stres 8036 f2ndres 8037 naddf 8717 ecopover 8859 xpf1o 9177 xpwdomg 9622 rankxplim 9916 imasaddfnlem 17574 imasvscafn 17583 comfeq 17750 isssc 17867 isfuncd 17915 cofucl 17938 funcres2b 17947 evlfcl 18278 uncfcurf 18295 yonedalem3 18336 yonedainv 18337 efgval2 19756 srgfcl 20213 txbas 23590 hausdiag 23668 tx1stc 23673 txkgen 23675 xkococn 23683 cnmpt21 23694 xkoinjcn 23710 tmdcn2 24112 clssubg 24132 qustgplem 24144 txmetcnp 24575 txmetcn 24576 qtopbaslem 24794 bndth 25003 cxpcn3 26805 mpodvdsmulf1o 27251 fsumdvdsmul 27252 dvdsmulf1o 27253 fsumdvdsmulOLD 27254 addsf 28029 xrofsup 32777 txpconn 35216 cvmlift2lem1 35286 cvmlift2lem12 35298 mclsax 35553 ismtyhmeolem 37790 dih1dimatlem 41311 ffnaov 47148 ovn0ssdmfun 48002 plusfreseq 48007 funcf2lem 48810 |
Copyright terms: Public domain | W3C validator |