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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 3567 | . 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 3565 | . 2 ⊢ (𝜑 → if(𝜒, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷)) |
6 | 2, 5 | eqtrd 2220 | 1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1363 ifcif 3546 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-rab 2474 df-v 2751 df-un 3145 df-if 3547 |
This theorem is referenced by: updjudhcoinlf 7093 updjudhcoinrg 7094 omp1eom 7108 xaddval 9859 iseqf1olemqval 10501 iseqf1olemqk 10508 seq3f1olemqsum 10514 exp3val 10536 cvgratz 11554 eucalgval2 12067 ennnfonelemg 12418 ennnfonelem1 12422 mulgval 13017 lgsval 14701 |
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