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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 3556 | . 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 3554 | . 2 ⊢ (𝜑 → if(𝜒, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷)) |
6 | 2, 5 | eqtrd 2210 | 1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1353 ifcif 3535 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rab 2464 df-v 2740 df-un 3134 df-if 3536 |
This theorem is referenced by: updjudhcoinlf 7079 updjudhcoinrg 7080 omp1eom 7094 xaddval 9845 iseqf1olemqval 10487 iseqf1olemqk 10494 seq3f1olemqsum 10500 exp3val 10522 cvgratz 11540 eucalgval2 12053 ennnfonelemg 12404 ennnfonelem1 12408 mulgval 12986 lgsval 14408 |
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