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Mirrors > Home > ILE Home > Th. List > ifeq2 | 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 2551 | . . 3 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} = {𝑥 ∈ 𝐵 ∣ ¬ 𝜑}) | |
2 | 1 | uneq2d 3097 | . 2 ⊢ (𝐴 = 𝐵 → ({𝑥 ∈ 𝐶 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = ({𝑥 ∈ 𝐶 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑})) |
3 | dfif6 3333 | . 2 ⊢ if(𝜑, 𝐶, 𝐴) = ({𝑥 ∈ 𝐶 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) | |
4 | dfif6 3333 | . 2 ⊢ if(𝜑, 𝐶, 𝐵) = ({𝑥 ∈ 𝐶 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑}) | |
5 | 2, 3, 4 | 3eqtr4g 2097 | 1 ⊢ (𝐴 = 𝐵 → if(𝜑, 𝐶, 𝐴) = if(𝜑, 𝐶, 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1243 {crab 2310 ∪ cun 2915 ifcif 3331 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rab 2315 df-v 2559 df-un 2922 df-if 3332 |
This theorem is referenced by: ifeq12 3344 ifeq2d 3346 ifbieq2i 3351 |
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