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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 2375. See cbviinvg 5048 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 2825 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶)) |
3 | 2 | cbvralvw 3235 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑧 ∈ 𝐶) |
4 | 3 | abbii 2807 | . 2 ⊢ {𝑧 ∣ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} = {𝑧 ∣ ∀𝑦 ∈ 𝐴 𝑧 ∈ 𝐶} |
5 | df-iin 4999 | . 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} | |
6 | df-iin 4999 | . 2 ⊢ ∩ 𝑦 ∈ 𝐴 𝐶 = {𝑧 ∣ ∀𝑦 ∈ 𝐴 𝑧 ∈ 𝐶} | |
7 | 4, 5, 6 | 3eqtr4i 2773 | 1 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑦 ∈ 𝐴 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 {cab 2712 ∀wral 3059 ∩ ciin 4997 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-iin 4999 |
This theorem is referenced by: meaiininc 46443 iinhoiicc 46630 smflimlem3 46729 smflimlem4 46730 smflimlem6 46732 smfsuplem2 46768 smflimsuplem1 46776 smflimsup 46784 |
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