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Mirrors > Home > MPE Home > Th. List > cbviinv | Structured version Visualization version GIF version |
Description: Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) Add disjoint variable condition to avoid ax-13 2370. See cbviinvg 5046 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.) |
Ref | Expression |
---|---|
cbviunv.1 | ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
cbviinv | ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑦 ∈ 𝐴 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2902 | . 2 ⊢ Ⅎ𝑦𝐵 | |
2 | nfcv 2902 | . 2 ⊢ Ⅎ𝑥𝐶 | |
3 | cbviunv.1 | . 2 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
4 | 1, 2, 3 | cbviin 5040 | 1 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑦 ∈ 𝐴 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∩ ciin 4998 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-11 2153 ax-12 2170 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1781 df-nf 1785 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-iin 5000 |
This theorem is referenced by: meaiininc 45662 iinhoiicc 45849 smflimlem3 45948 smflimlem4 45949 smflimlem6 45951 smfsuplem2 45987 smflimsuplem1 45995 smflimsup 46003 |
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