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| Mirrors > Home > MPE Home > Th. List > cbviin | Structured version Visualization version GIF version | ||
| Description: Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) (Revised by Mario Carneiro, 14-Oct-2016.) Add disjoint variable condition to avoid ax-13 2377. See cbviing 5039 for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024.) |
| Ref | Expression |
|---|---|
| cbviun.1 | ⊢ Ⅎ𝑦𝐵 |
| cbviun.2 | ⊢ Ⅎ𝑥𝐶 |
| cbviun.3 | ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) |
| Ref | Expression |
|---|---|
| cbviin | ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑦 ∈ 𝐴 𝐶 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cbviun.1 | . . . . 5 ⊢ Ⅎ𝑦𝐵 | |
| 2 | 1 | nfcri 2897 | . . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵 |
| 3 | cbviun.2 | . . . . 5 ⊢ Ⅎ𝑥𝐶 | |
| 4 | 3 | nfcri 2897 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐶 |
| 5 | cbviun.3 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
| 6 | 5 | eleq2d 2827 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶)) |
| 7 | 2, 4, 6 | cbvralw 3306 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑧 ∈ 𝐶) |
| 8 | 7 | abbii 2809 | . 2 ⊢ {𝑧 ∣ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} = {𝑧 ∣ ∀𝑦 ∈ 𝐴 𝑧 ∈ 𝐶} |
| 9 | df-iin 4994 | . 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} | |
| 10 | df-iin 4994 | . 2 ⊢ ∩ 𝑦 ∈ 𝐴 𝐶 = {𝑧 ∣ ∀𝑦 ∈ 𝐴 𝑧 ∈ 𝐶} | |
| 11 | 8, 9, 10 | 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 Ⅎwnfc 2890 ∀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-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-iin 4994 |
| This theorem is referenced by: elrfirn2 42707 fnlimfvre 45689 smflimlem6 46791 smflim 46792 smflim2 46821 smfsup 46829 smfinflem 46832 smfinf 46833 smflimsup 46843 smfliminf 46846 |
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