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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 8121 | . 2 ⊢ (𝜑 → (𝐵 + 𝐶) = (𝐶 + 𝐵)) |
5 | 1, 4 | eqtrd 2220 | 1 ⊢ (𝜑 → 𝐴 = (𝐶 + 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1363 ∈ wcel 2158 (class class class)co 5888 ℂcc 7822 + caddc 7827 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1457 ax-gen 1459 ax-4 1520 ax-17 1536 ax-ext 2169 ax-addcom 7924 |
This theorem depends on definitions: df-bi 117 df-cleq 2180 |
This theorem is referenced by: mvrladdd 8337 hashfz 10814 bdtrilem 11260 clim2ser2 11359 fsumparts 11491 arisum 11519 divalglemnn 11936 phiprmpw 12235 mulgdir 13044 metrtri 14117 apdifflemr 15036 |
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