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Mirrors > Home > MPE Home > Th. List > ifeq2da | Structured version Visualization version GIF version |
Description: Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
ifeq2da.1 | ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
ifeq2da | ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iftrue 4535 | . . . 4 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐴) = 𝐶) | |
2 | iftrue 4535 | . . . 4 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐵) = 𝐶) | |
3 | 1, 2 | eqtr4d 2776 | . . 3 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)) |
4 | 3 | adantl 483 | . 2 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)) |
5 | ifeq2da.1 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐴 = 𝐵) | |
6 | 5 | ifeq2d 4549 | . 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)) |
7 | 4, 6 | pm2.61dan 812 | 1 ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ifcif 4529 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-un 3954 df-if 4530 |
This theorem is referenced by: ifeq12da 4562 dfac12lem1 10138 ttukeylem3 10506 xmulcom 13245 xmulneg1 13248 subgmulg 19020 1marepvmarrepid 22077 pcopt2 24539 limcdif 25393 limcmpt 25400 limcres 25403 limccnp 25408 radcnv0 25928 leibpi 26447 efrlim 26474 dchrvmasumiflem2 27005 padicabvf 27134 padicabvcxp 27135 itg2addnclem 36539 fourierdlem73 44895 fourierdlem76 44898 fourierdlem89 44911 fourierdlem91 44913 |
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