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| Mirrors > Home > MPE Home > Th. List > ifeq2 | 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 |
|---|---|
| ifeq2 | ⊢ (𝐴 = 𝐵 → if(𝜑, 𝐶, 𝐴) = if(𝜑, 𝐶, 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rabeq 3451 | . . 3 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} = {𝑥 ∈ 𝐵 ∣ ¬ 𝜑}) | |
| 2 | 1 | uneq2d 4168 | . 2 ⊢ (𝐴 = 𝐵 → ({𝑥 ∈ 𝐶 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = ({𝑥 ∈ 𝐶 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑})) |
| 3 | dfif6 4528 | . 2 ⊢ if(𝜑, 𝐶, 𝐴) = ({𝑥 ∈ 𝐶 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) | |
| 4 | dfif6 4528 | . 2 ⊢ if(𝜑, 𝐶, 𝐵) = ({𝑥 ∈ 𝐶 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑}) | |
| 5 | 2, 3, 4 | 3eqtr4g 2802 | 1 ⊢ (𝐴 = 𝐵 → if(𝜑, 𝐶, 𝐴) = if(𝜑, 𝐶, 𝐵)) |
| 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 ifeq2d 4546 ifbieq2i 4551 somincom 6154 mdetunilem9 22626 prmorcht 27221 pclogsum 27259 matunitlindflem1 37623 ftc1anclem6 37705 ftc1anclem8 37707 ftc1anc 37708 hdmap1cbv 41804 reabssgn 43649 hoidmv1le 46609 hoidmvlelem3 46612 vonn0ioo 46702 vonn0icc 46703 |
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