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| 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 5758 | . . 3 ⊢ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) = (𝐴 × 𝐵) | |
| 2 | 1 | raleqi 3324 | . 2 ⊢ (∀𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∀𝑥 ∈ (𝐴 × 𝐵)𝜑) |
| 3 | ralxp.1 | . . 3 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → (𝜑 ↔ 𝜓)) | |
| 4 | 3 | raliunxp 5850 | . 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 1540 ∀wral 3061 {csn 4626 〈cop 4632 ∪ ciun 4991 × cxp 5683 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-iun 4993 df-opab 5206 df-xp 5691 df-rel 5692 |
| This theorem is referenced by: ralxpf 5857 reu3op 6312 f1opr 7489 ffnov 7559 eqfnov 7562 funimassov 7610 f1stres 8038 f2ndres 8039 naddf 8719 ecopover 8861 xpf1o 9179 xpwdomg 9625 rankxplim 9919 imasaddfnlem 17573 imasvscafn 17582 comfeq 17749 isssc 17864 isfuncd 17910 cofucl 17933 funcres2b 17942 evlfcl 18267 uncfcurf 18284 yonedalem3 18325 yonedainv 18326 efgval2 19742 srgfcl 20193 txbas 23575 hausdiag 23653 tx1stc 23658 txkgen 23660 xkococn 23668 cnmpt21 23679 xkoinjcn 23695 tmdcn2 24097 clssubg 24117 qustgplem 24129 txmetcnp 24560 txmetcn 24561 qtopbaslem 24779 bndth 24990 cxpcn3 26791 mpodvdsmulf1o 27237 fsumdvdsmul 27238 dvdsmulf1o 27239 fsumdvdsmulOLD 27240 addsf 28015 xrofsup 32771 txpconn 35237 cvmlift2lem1 35307 cvmlift2lem12 35319 mclsax 35574 ismtyhmeolem 37811 dih1dimatlem 41331 ffnaov 47211 ovn0ssdmfun 48075 plusfreseq 48080 funcf2lem 48914 fucofulem2 49006 |
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