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| Mirrors > Home > MPE Home > Th. List > iineq1 | Structured version Visualization version GIF version | ||
| Description: Equality theorem for indexed intersection. (Contributed by NM, 27-Jun-1998.) |
| Ref | Expression |
|---|---|
| iineq1 | ⊢ (𝐴 = 𝐵 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | raleq 3323 | . . 3 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐶)) | |
| 2 | 1 | abbidv 2808 | . 2 ⊢ (𝐴 = 𝐵 → {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶} = {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐶}) |
| 3 | df-iin 4994 | . 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶} | |
| 4 | df-iin 4994 | . 2 ⊢ ∩ 𝑥 ∈ 𝐵 𝐶 = {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐶} | |
| 5 | 2, 3, 4 | 3eqtr4g 2802 | 1 ⊢ (𝐴 = 𝐵 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 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-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-ral 3062 df-rex 3071 df-iin 4994 |
| This theorem is referenced by: iinrab2 5070 iinvdif 5080 riin0 5082 iin0 5362 xpriindi 5847 cmpfi 23416 ptbasfi 23589 fclsval 24016 taylfval 26400 polvalN 39907 iineq1d 45095 |
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