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Mirrors > Home > ILE Home > Th. List > ralxp | GIF version |
Description: Universal quantification restricted to a cross 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 4400 | . . 3 ⊢ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) = (𝐴 × 𝐵) | |
2 | 1 | raleqi 2509 | . 2 ⊢ (∀𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∀𝑥 ∈ (𝐴 × 𝐵)𝜑) |
3 | ralxp.1 | . . 3 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → (𝜑 ↔ 𝜓)) | |
4 | 3 | raliunxp 4477 | . 2 ⊢ (∀𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜓) |
5 | 2, 4 | bitr3i 175 | 1 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1243 ∀wral 2306 {csn 3375 〈cop 3378 ∪ ciun 3657 × cxp 4343 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-csb 2853 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-iun 3659 df-opab 3819 df-xp 4351 df-rel 4352 |
This theorem is referenced by: ralxpf 4482 issref 4707 ffnov 5605 eqfnov 5607 funimassov 5650 f1stres 5786 f2ndres 5787 ecopover 6204 ecopoverg 6207 |
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