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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 2974 | . 2 ⊢ Ⅎ𝑦𝐵 | |
2 | nfcv 2974 | . 2 ⊢ Ⅎ𝑥𝐶 | |
3 | cbvixpv.1 | . 2 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
4 | 1, 2, 3 | cbvixp 8466 | 1 ⊢ X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 Xcixp 8449 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fn 6351 df-fv 6356 df-ixp 8450 |
This theorem is referenced by: funcpropd 17158 invfuc 17232 natpropd 17234 dprdw 19061 dprdwd 19062 ptuni2 22112 ptbasin 22113 ptbasfi 22117 ptpjopn 22148 ptclsg 22151 dfac14 22154 ptcnp 22158 ptcmplem2 22589 ptcmpg 22593 prdsxmslem2 23066 upixp 34885 rrxsnicc 42462 ioorrnopn 42467 ioorrnopnxr 42469 ovnsubadd 42731 hoidmvlelem4 42757 hoidmvle 42759 hspdifhsp 42775 hoiqssbllem2 42782 hspmbl 42788 hoimbl 42790 opnvonmbl 42793 ovnovollem3 42817 |
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