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Mirrors > Home > ILE Home > Th. List > ifeq12d | GIF version |
Description: Equality deduction for conditional operator. (Contributed by NM, 24-Mar-2015.) |
Ref | Expression |
---|---|
ifeq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
ifeq12d.2 | ⊢ (𝜑 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
ifeq12d | ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifeq1d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | 1 | ifeq1d 3345 | . 2 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶)) |
3 | ifeq12d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
4 | 3 | ifeq2d 3346 | . 2 ⊢ (𝜑 → if(𝜓, 𝐵, 𝐶) = if(𝜓, 𝐵, 𝐷)) |
5 | 2, 4 | eqtrd 2072 | 1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 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: ifbieq12d 3354 expival 9257 |
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