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| Mirrors > Home > MPE Home > Th. List > ifeq1 | Structured version Visualization version GIF version | ||
| Description: Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
| Ref | Expression |
|---|---|
| ifeq1 | ⊢ (𝐴 = 𝐵 → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐶)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rabeq 3451 | . . 3 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}) | |
| 2 | 1 | uneq1d 4167 | . 2 ⊢ (𝐴 = 𝐵 → ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐶 ∣ ¬ 𝜑}) = ({𝑥 ∈ 𝐵 ∣ 𝜑} ∪ {𝑥 ∈ 𝐶 ∣ ¬ 𝜑})) |
| 3 | dfif6 4528 | . 2 ⊢ if(𝜑, 𝐴, 𝐶) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐶 ∣ ¬ 𝜑}) | |
| 4 | dfif6 4528 | . 2 ⊢ if(𝜑, 𝐵, 𝐶) = ({𝑥 ∈ 𝐵 ∣ 𝜑} ∪ {𝑥 ∈ 𝐶 ∣ ¬ 𝜑}) | |
| 5 | 2, 3, 4 | 3eqtr4g 2802 | 1 ⊢ (𝐴 = 𝐵 → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 {crab 3436 ∪ cun 3949 ifcif 4525 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-un 3956 df-if 4526 |
| This theorem is referenced by: ifeq12 4544 ifeq1d 4545 ifbieq12i 4553 rdgeq2 8452 dfoi 9551 wemaplem2 9587 cantnflem1 9729 prodeq2w 15946 prodeq2ii 15947 mgm2nsgrplem2 18932 mgm2nsgrplem3 18933 mplcoe3 22056 marrepval0 22567 ellimc 25908 ply1nzb 26162 dchrvmasumiflem1 27545 signspval 34567 dfrdg2 35796 sumeq2si 36203 prodeq2si 36205 dfafv22 47271 |
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