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Mirrors > Home > ILE Home > Th. List > ifbieq12d | GIF version |
Description: Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
ifbieq12d.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
ifbieq12d.2 | ⊢ (𝜑 → 𝐴 = 𝐶) |
ifbieq12d.3 | ⊢ (𝜑 → 𝐵 = 𝐷) |
Ref | Expression |
---|---|
ifbieq12d | ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifbieq12d.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | 1 | ifbid 3498 | . 2 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐴, 𝐵)) |
3 | ifbieq12d.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐶) | |
4 | ifbieq12d.3 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐷) | |
5 | 3, 4 | ifeq12d 3496 | . 2 ⊢ (𝜑 → if(𝜒, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷)) |
6 | 2, 5 | eqtrd 2173 | 1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷)) |
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: updjudhcoinlf 6973 updjudhcoinrg 6974 omp1eom 6988 xaddval 9658 iseqf1olemqval 10291 iseqf1olemqk 10298 seq3f1olemqsum 10304 exp3val 10326 cvgratz 11333 eucalgval2 11770 ennnfonelemg 11952 ennnfonelem1 11956 ressid2 12057 ressval2 12058 |
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