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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 2377. See cbviinvg 5043 for a less restrictive version requiring more axioms. (Revised by GG, 14-Aug-2025.) | 
| Ref | Expression | 
|---|---|
| cbviunv.1 | ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | 
| Ref | Expression | 
|---|---|
| cbviinv | ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑦 ∈ 𝐴 𝐶 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | cbviunv.1 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
| 2 | 1 | eleq2d 2827 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶)) | 
| 3 | 2 | cbvralvw 3237 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑧 ∈ 𝐶) | 
| 4 | 3 | abbii 2809 | . 2 ⊢ {𝑧 ∣ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} = {𝑧 ∣ ∀𝑦 ∈ 𝐴 𝑧 ∈ 𝐶} | 
| 5 | df-iin 4994 | . 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} | |
| 6 | df-iin 4994 | . 2 ⊢ ∩ 𝑦 ∈ 𝐴 𝐶 = {𝑧 ∣ ∀𝑦 ∈ 𝐴 𝑧 ∈ 𝐶} | |
| 7 | 4, 5, 6 | 3eqtr4i 2775 | 1 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑦 ∈ 𝐴 𝐶 | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {cab 2714 ∀wral 3061 ∩ ciin 4992 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-iin 4994 | 
| This theorem is referenced by: meaiininc 46502 iinhoiicc 46689 smflimlem3 46788 smflimlem4 46789 smflimlem6 46791 smfsuplem2 46827 smflimsuplem1 46835 smflimsup 46843 | 
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