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Mirrors > Home > ILE Home > Th. List > comraddd | GIF version |
Description: Commute RHS addition, in deduction form. (Contributed by David A. Wheeler, 11-Oct-2018.) |
Ref | Expression |
---|---|
comraddd.1 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
comraddd.2 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
comraddd.3 | ⊢ (𝜑 → 𝐴 = (𝐵 + 𝐶)) |
Ref | Expression |
---|---|
comraddd | ⊢ (𝜑 → 𝐴 = (𝐶 + 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | comraddd.3 | . 2 ⊢ (𝜑 → 𝐴 = (𝐵 + 𝐶)) | |
2 | comraddd.1 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | comraddd.2 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
4 | 2, 3 | addcomd 8070 | . 2 ⊢ (𝜑 → (𝐵 + 𝐶) = (𝐶 + 𝐵)) |
5 | 1, 4 | eqtrd 2203 | 1 ⊢ (𝜑 → 𝐴 = (𝐶 + 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ∈ wcel 2141 (class class class)co 5853 ℂcc 7772 + caddc 7777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1440 ax-gen 1442 ax-4 1503 ax-17 1519 ax-ext 2152 ax-addcom 7874 |
This theorem depends on definitions: df-bi 116 df-cleq 2163 |
This theorem is referenced by: mvrladdd 8286 hashfz 10756 bdtrilem 11202 clim2ser2 11301 fsumparts 11433 arisum 11461 divalglemnn 11877 phiprmpw 12176 metrtri 13171 apdifflemr 14079 |
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