Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > ifbieq2d | GIF version |
Description: Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.) |
Ref | Expression |
---|---|
ifbieq2d.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
ifbieq2d.2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
ifbieq2d | ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜒, 𝐶, 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifbieq2d.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | 1 | ifbid 3493 | . 2 ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜒, 𝐶, 𝐴)) |
3 | ifbieq2d.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
4 | 3 | ifeq2d 3490 | . 2 ⊢ (𝜑 → if(𝜒, 𝐶, 𝐴) = if(𝜒, 𝐶, 𝐵)) |
5 | 2, 4 | eqtrd 2172 | 1 ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜒, 𝐶, 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1331 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-in1 603 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-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 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-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rab 2425 df-v 2688 df-un 3075 df-if 3475 |
This theorem is referenced by: difinfsnlem 6984 ctmlemr 6993 xnegeq 9610 xaddval 9628 iseqf1olemqval 10260 iseqf1olemqk 10267 seq3f1olemqsum 10273 exp3val 10295 gcdval 11648 gcdass 11703 lcmval 11744 lcmass 11766 ennnfonelemj0 11914 ennnfonelemjn 11915 ennnfonelem0 11918 ennnfonelemp1 11919 ennnfonelemnn0 11935 nnsf 13199 peano4nninf 13200 peano3nninf 13201 exmidsbthr 13218 |
Copyright terms: Public domain | W3C validator |