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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumeq12dva | Structured version Visualization version GIF version |
Description: Equality deduction for extended sum. (Contributed by Thierry Arnoux, 18-Feb-2017.) (Revised by Thierry Arnoux, 29-Jun-2017.) |
Ref | Expression |
---|---|
esumeq12dva.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
esumeq12dva.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
esumeq12dva | ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1917 | . 2 ⊢ Ⅎ𝑘𝜑 | |
2 | esumeq12dva.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
3 | esumeq12dva.2 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 = 𝐷) | |
4 | 1, 2, 3 | esumeq12dvaf 32753 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Σ*cesum 32749 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rab 3419 df-v 3461 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-nul 4303 df-if 4507 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-br 5126 df-opab 5188 df-mpt 5209 df-iota 6468 df-fv 6524 df-ov 7380 df-esum 32750 |
This theorem is referenced by: esumeq12d 32755 |
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