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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumeq12d | Structured version Visualization version GIF version |
Description: Equality deduction for extended sum. (Contributed by Thierry Arnoux, 18-Feb-2017.) |
Ref | Expression |
---|---|
esumeq12d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
esumeq12d.2 | ⊢ (𝜑 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
esumeq12d | ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | esumeq12d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | esumeq12d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
3 | 2 | adantr 473 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 = 𝐷) |
4 | 1, 3 | esumeq12dva 30935 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 Σ*cesum 30930 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2750 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3417 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-nul 4181 df-if 4352 df-sn 4443 df-pr 4445 df-op 4449 df-uni 4714 df-br 4931 df-opab 4993 df-mpt 5010 df-iota 6154 df-fv 6198 df-ov 6981 df-esum 30931 |
This theorem is referenced by: esumeq1 30937 esum2dlem 30995 |
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