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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.) |
Ref | Expression |
---|---|
cbviunv.1 | ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
cbviinv | ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑦 ∈ 𝐴 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2932 | . 2 ⊢ Ⅎ𝑦𝐵 | |
2 | nfcv 2932 | . 2 ⊢ Ⅎ𝑥𝐶 | |
3 | cbviunv.1 | . 2 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
4 | 1, 2, 3 | cbviin 4832 | 1 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑦 ∈ 𝐴 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∩ ciin 4793 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ral 3093 df-iin 4795 |
This theorem is referenced by: meaiininc 42206 iinhoiicc 42393 smflimlem3 42486 smflimlem4 42487 smflimlem6 42489 smfsuplem2 42523 smflimsuplem1 42531 smflimsup 42539 |
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