![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4537 | . . . 4 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐴) = 𝐶) | |
2 | iftrue 4537 | . . . 4 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐵) = 𝐶) | |
3 | 1, 2 | eqtr4d 2778 | . . 3 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)) |
4 | 3 | adantl 481 | . 2 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)) |
5 | ifeq2da.1 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐴 = 𝐵) | |
6 | 5 | ifeq2d 4551 | . 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)) |
7 | 4, 6 | pm2.61dan 813 | 1 ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ifcif 4531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-un 3968 df-if 4532 |
This theorem is referenced by: ifeq12da 4564 dfac12lem1 10182 ttukeylem3 10549 xmulcom 13305 xmulneg1 13308 subgmulg 19171 1marepvmarrepid 22597 pcopt2 25070 limcdif 25926 limcmpt 25933 limcres 25936 limccnp 25941 radcnv0 26474 leibpi 27000 efrlim 27027 efrlimOLD 27028 dchrvmasumiflem2 27561 padicabvf 27690 padicabvcxp 27691 itg2addnclem 37658 fourierdlem73 46135 fourierdlem76 46138 fourierdlem89 46151 fourierdlem91 46153 |
Copyright terms: Public domain | W3C validator |