Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2381. See cbviinvg 4959 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 2974 | . 2 ⊢ Ⅎ𝑦𝐵 | |
2 | nfcv 2974 | . 2 ⊢ Ⅎ𝑥𝐶 | |
3 | cbviunv.1 | . 2 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
4 | 1, 2, 3 | cbviin 4953 | 1 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑦 ∈ 𝐴 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∩ ciin 4911 |
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-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-iin 4913 |
This theorem is referenced by: meaiininc 42646 iinhoiicc 42833 smflimlem3 42926 smflimlem4 42927 smflimlem6 42929 smfsuplem2 42963 smflimsuplem1 42971 smflimsup 42979 |
Copyright terms: Public domain | W3C validator |