| Mathbox for Glauco Siliprandi |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > iineq1d | Structured version Visualization version GIF version | ||
| Description: Equality theorem for indexed intersection. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
| Ref | Expression |
|---|---|
| iineq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
| Ref | Expression |
|---|---|
| iineq1d | ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | iineq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 2 | iineq1 5009 | . 2 ⊢ (𝐴 = 𝐵 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶) | |
| 3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∩ 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: smflimlem2 46787 smflimlem3 46788 smflimlem4 46789 smflim2 46821 smflimsuplem1 46835 smflimsuplem7 46841 smflimsup 46843 smfliminf 46846 |
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