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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 3570 | . 2 ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜒, 𝐶, 𝐴)) |
3 | ifbieq2d.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
4 | 3 | ifeq2d 3567 | . 2 ⊢ (𝜑 → if(𝜒, 𝐶, 𝐴) = if(𝜒, 𝐶, 𝐵)) |
5 | 2, 4 | eqtrd 2222 | 1 ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜒, 𝐶, 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1364 ifcif 3549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-rab 2477 df-v 2754 df-un 3148 df-if 3550 |
This theorem is referenced by: difinfsnlem 7128 ctmlemr 7137 xnegeq 9857 xaddval 9875 iseqf1olemqval 10518 iseqf1olemqk 10525 seq3f1olemqsum 10531 exp3val 10553 gcdval 11992 gcdass 12048 lcmval 12095 lcmass 12117 pcval 12328 ennnfonelemj0 12452 ennnfonelemjn 12453 ennnfonelem0 12456 ennnfonelemp1 12457 ennnfonelemnn0 12473 mulgval 13064 znval 13932 lgsval 14863 lgsfvalg 14864 lgsval2lem 14869 nnsf 15213 peano4nninf 15214 peano3nninf 15215 exmidsbthr 15230 |
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