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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 1909 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | iuneq2dv.1 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) | |
3 | 1, 2 | iineq2d 4933 | 1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ∩ ciin 4911 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-12 2170 ax-ext 2791 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1775 df-nf 1779 df-sb 2064 df-clab 2798 df-cleq 2812 df-clel 2891 df-ral 3141 df-iin 4913 |
This theorem is referenced by: cntziinsn 18457 ptbasfi 22181 fclsval 22608 taylfval 24939 polfvalN 37032 dihglblem3N 38423 dihmeetlem2N 38427 iineq12dv 41363 saliincl 42601 iccvonmbllem 42951 vonicclem2 42957 smflimlem3 43040 smflimlem4 43041 smflimlem6 43043 smflimsuplem3 43087 |
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