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Mirrors > Home > MPE Home > Th. List > ifeq12 | Structured version Visualization version GIF version |
Description: Equality theorem for conditional operators. (Contributed by NM, 1-Sep-2004.) |
Ref | Expression |
---|---|
ifeq12 | ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifeq1 4528 | . 2 ⊢ (𝐴 = 𝐵 → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐶)) | |
2 | ifeq2 4529 | . 2 ⊢ (𝐶 = 𝐷 → if(𝜑, 𝐵, 𝐶) = if(𝜑, 𝐵, 𝐷)) | |
3 | 1, 2 | sylan9eq 2787 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ifcif 4524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-rab 3428 df-v 3471 df-un 3949 df-if 4525 |
This theorem is referenced by: xaddmnf1 13231 xpsrnbas 17544 ditg0 25769 mumullem2 27099 sqrtcval 42994 sqrtcval2 42995 |
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