Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > ifcldadc | GIF version |
Description: Conditional closure. (Contributed by Jim Kingdon, 11-Jan-2022.) |
Ref | Expression |
---|---|
ifcldadc.1 | ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ 𝐶) |
ifcldadc.2 | ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 ∈ 𝐶) |
ifcldadc.dc | ⊢ (𝜑 → DECID 𝜓) |
Ref | Expression |
---|---|
ifcldadc | ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iftrue 3479 | . . . 4 ⊢ (𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐴) | |
2 | 1 | adantl 275 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐴, 𝐵) = 𝐴) |
3 | ifcldadc.1 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ 𝐶) | |
4 | 2, 3 | eqeltrd 2216 | . 2 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐴, 𝐵) ∈ 𝐶) |
5 | iffalse 3482 | . . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐵) | |
6 | 5 | adantl 275 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐵) = 𝐵) |
7 | ifcldadc.2 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 ∈ 𝐶) | |
8 | 6, 7 | eqeltrd 2216 | . 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐵) ∈ 𝐶) |
9 | ifcldadc.dc | . . 3 ⊢ (𝜑 → DECID 𝜓) | |
10 | exmiddc 821 | . . 3 ⊢ (DECID 𝜓 → (𝜓 ∨ ¬ 𝜓)) | |
11 | 9, 10 | syl 14 | . 2 ⊢ (𝜑 → (𝜓 ∨ ¬ 𝜓)) |
12 | 4, 8, 11 | mpjaodan 787 | 1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 697 DECID wdc 819 = wceq 1331 ∈ wcel 1480 ifcif 3474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-11 1484 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-if 3475 |
This theorem is referenced by: updjudhf 6964 omp1eomlem 6979 difinfsnlem 6984 ctmlemr 6993 ctssdclemn0 6995 ctssdc 6998 enumctlemm 6999 xaddf 9627 xaddval 9628 iseqf1olemqcl 10259 iseqf1olemnab 10261 iseqf1olemjpcl 10268 iseqf1olemqpcl 10269 seq3f1oleml 10276 seq3f1o 10277 exp3val 10295 xrmaxiflemcl 11014 summodclem2a 11150 zsumdc 11153 fsum3 11156 isumss 11160 fsum3cvg2 11163 fsum3ser 11166 fsumcl2lem 11167 fsumadd 11175 sumsnf 11178 sumsplitdc 11201 fsummulc2 11217 isumlessdc 11265 cvgratz 11301 prodmodclem3 11344 prodmodclem2a 11345 eucalgval2 11734 lcmval 11744 ennnfonelemg 11916 subctctexmid 13196 |
Copyright terms: Public domain | W3C validator |