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Mirrors > Home > MPE Home > Th. List > iineq2dv | Structured version Visualization version GIF version |
Description: Equality deduction for indexed intersection. (Contributed by NM, 3-Aug-2004.) |
Ref | Expression |
---|---|
iuneq2dv.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
iineq2dv | ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1915 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | iuneq2dv.1 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) | |
3 | 1, 2 | iineq2d 4904 | 1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∩ ciin 4882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ral 3111 df-iin 4884 |
This theorem is referenced by: cntziinsn 18457 ptbasfi 22186 fclsval 22613 taylfval 24954 polfvalN 37200 dihglblem3N 38591 dihmeetlem2N 38595 iineq12dv 41742 saliincl 42967 iccvonmbllem 43317 vonicclem2 43323 smflimlem3 43406 smflimlem4 43407 smflimlem6 43409 smflimsuplem3 43453 |
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