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Mirrors > Home > MPE Home > Th. List > cbvixpv | Structured version Visualization version GIF version |
Description: Change bound variable in an indexed Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
cbvixpv.1 | ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
cbvixpv | ⊢ X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2751 | . 2 ⊢ Ⅎ𝑦𝐵 | |
2 | nfcv 2751 | . 2 ⊢ Ⅎ𝑥𝐶 | |
3 | cbvixpv.1 | . 2 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
4 | 1, 2, 3 | cbvixp 7811 | 1 ⊢ X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 Xcixp 7794 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fn 5807 df-fv 5812 df-ixp 7795 |
This theorem is referenced by: funcpropd 16383 invfuc 16457 natpropd 16459 dprdw 18232 dprdwd 18233 ptuni2 21189 ptbasin 21190 ptbasfi 21194 ptpjopn 21225 ptclsg 21228 dfac14 21231 ptcnp 21235 ptcmplem2 21667 ptcmpg 21671 prdsxmslem2 22144 upixp 32694 rrxsnicc 39196 ioorrnopn 39201 ioorrnopnxr 39203 ovnsubadd 39462 hoidmvlelem4 39488 hoidmvle 39490 hspdifhsp 39506 hoiqssbllem2 39513 hspmbl 39519 hoimbl 39521 opnvonmbl 39524 ovnovollem3 39548 |
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