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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 4429 | . 2 ⊢ (𝐴 = 𝐵 → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐶)) | |
2 | ifeq2 4430 | . 2 ⊢ (𝐶 = 𝐷 → if(𝜑, 𝐵, 𝐶) = if(𝜑, 𝐵, 𝐷)) | |
3 | 1, 2 | sylan9eq 2853 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ifcif 4425 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-rab 3115 df-v 3443 df-un 3886 df-if 4426 |
This theorem is referenced by: xaddmnf1 12609 xpsrnbas 16836 ditg0 24456 mumullem2 25765 sqrtcval 40341 sqrtcval2 40342 |
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