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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 3412 | . 2 ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜒, 𝐶, 𝐴)) |
3 | ifbieq2d.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
4 | 3 | ifeq2d 3409 | . 2 ⊢ (𝜑 → if(𝜒, 𝐶, 𝐴) = if(𝜒, 𝐶, 𝐵)) |
5 | 2, 4 | eqtrd 2120 | 1 ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜒, 𝐶, 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 = wceq 1289 ifcif 3393 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-rab 2368 df-v 2621 df-un 3003 df-if 3394 |
This theorem is referenced by: xnegeq 9289 iseqf1olemqval 9916 iseqf1olemqk 9923 seq3f1olemqsum 9929 exp3val 9957 gcdval 11229 gcdass 11282 lcmval 11323 lcmass 11345 nnsf 11895 peano4nninf 11896 peano3nninf 11897 exmidsbthr 11913 |
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