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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumeq2sdv | Structured version Visualization version GIF version |
Description: Equality deduction for extended sum. (Contributed by Thierry Arnoux, 25-Dec-2016.) |
Ref | Expression |
---|---|
esumeq2sdv.1 | ⊢ (𝜑 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
esumeq2sdv | ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | esumeq2sdv.1 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐶) | |
2 | 1 | adantr 484 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 = 𝐶) |
3 | 2 | esumeq2dv 31690 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 Σ*cesum 31679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-ral 3059 df-rab 3063 df-v 3403 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-mpt 5125 df-iota 6327 df-fv 6377 df-ov 7205 df-esum 31680 |
This theorem is referenced by: ismeas 31851 isrnmeas 31852 |
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