Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iineq12dv | Structured version Visualization version GIF version |
Description: Equality deduction for indexed intersection. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
iineq12dv.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
iineq12dv.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
iineq12dv | ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iineq12dv.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | 1 | iineq1d 41376 | . 2 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶) |
3 | iineq12dv.2 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 = 𝐷) | |
4 | 3 | iineq2dv 4944 | . 2 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐵 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷) |
5 | 2, 4 | eqtrd 2856 | 1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∩ ciin 4920 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-ral 3143 df-iin 4922 |
This theorem is referenced by: smflim 43073 |
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