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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 3498 | . 2 ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜒, 𝐶, 𝐴)) |
3 | ifbieq2d.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
4 | 3 | ifeq2d 3495 | . 2 ⊢ (𝜑 → if(𝜒, 𝐶, 𝐴) = if(𝜒, 𝐶, 𝐵)) |
5 | 2, 4 | eqtrd 2173 | 1 ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜒, 𝐶, 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1332 ifcif 3479 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-rab 2426 df-v 2691 df-un 3080 df-if 3480 |
This theorem is referenced by: difinfsnlem 6992 ctmlemr 7001 xnegeq 9640 xaddval 9658 iseqf1olemqval 10291 iseqf1olemqk 10298 seq3f1olemqsum 10304 exp3val 10326 gcdval 11684 gcdass 11739 lcmval 11780 lcmass 11802 ennnfonelemj0 11950 ennnfonelemjn 11951 ennnfonelem0 11954 ennnfonelemp1 11955 ennnfonelemnn0 11971 nnsf 13374 peano4nninf 13375 peano3nninf 13376 exmidsbthr 13393 |
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