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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 3166 | . . 3 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}) | |
2 | 1 | uneq1d 3728 | . 2 ⊢ (𝐴 = 𝐵 → ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐶 ∣ ¬ 𝜑}) = ({𝑥 ∈ 𝐵 ∣ 𝜑} ∪ {𝑥 ∈ 𝐶 ∣ ¬ 𝜑})) |
3 | dfif6 4039 | . 2 ⊢ if(𝜑, 𝐴, 𝐶) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐶 ∣ ¬ 𝜑}) | |
4 | dfif6 4039 | . 2 ⊢ if(𝜑, 𝐵, 𝐶) = ({𝑥 ∈ 𝐵 ∣ 𝜑} ∪ {𝑥 ∈ 𝐶 ∣ ¬ 𝜑}) | |
5 | 2, 3, 4 | 3eqtr4g 2669 | 1 ⊢ (𝐴 = 𝐵 → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1475 {crab 2900 ∪ cun 3538 ifcif 4036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-un 3545 df-if 4037 |
This theorem is referenced by: ifeq12 4053 ifeq1d 4054 ifbieq12i 4062 ifexg 4107 rdgeq2 7395 dfoi 8299 wemaplem2 8335 cantnflem1 8469 prodeq2w 14481 prodeq2ii 14482 mgm2nsgrplem2 17229 mgm2nsgrplem3 17230 mplcoe3 19287 marrepval0 20186 ellimc 23443 ply1nzb 23686 dchrvmasumiflem1 24990 signspval 29955 dfrdg2 30945 dfafv2 39861 |
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