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Mirrors > Home > ILE Home > Th. List > rexxp | GIF version |
Description: Existential quantification restricted to a cross product is equivalent to a double restricted quantification. (Contributed by NM, 11-Nov-1995.) (Revised by Mario Carneiro, 14-Feb-2015.) |
Ref | Expression |
---|---|
ralxp.1 | ⊢ (x = 〈y, z〉 → (φ ↔ ψ)) |
Ref | Expression |
---|---|
rexxp | ⊢ (∃x ∈ (A × B)φ ↔ ∃y ∈ A ∃z ∈ B ψ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunxpconst 4343 | . . 3 ⊢ ∪ y ∈ A ({y} × B) = (A × B) | |
2 | 1 | rexeqi 2504 | . 2 ⊢ (∃x ∈ ∪ y ∈ A ({y} × B)φ ↔ ∃x ∈ (A × B)φ) |
3 | ralxp.1 | . . 3 ⊢ (x = 〈y, z〉 → (φ ↔ ψ)) | |
4 | 3 | rexiunxp 4421 | . 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 1242 ∃wrex 2301 {csn 3367 〈cop 3370 ∪ ciun 3648 × cxp 4286 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-sbc 2759 df-csb 2847 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-iun 3650 df-opab 3810 df-xp 4294 df-rel 4295 |
This theorem is referenced by: rexxpf 4426 fnrnov 5588 foov 5589 ovelimab 5593 cnref1o 8357 |
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