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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 3416 | . 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 3414 | . 2 ⊢ (𝜑 → if(𝜒, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷)) |
6 | 2, 5 | eqtrd 2121 | 1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1290 ifcif 3397 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-rab 2369 df-v 2622 df-un 3004 df-if 3398 |
This theorem is referenced by: updjudhcoinlf 6825 updjudhcoinrg 6826 iseqf1olemqval 9970 iseqf1olemqk 9977 seq3f1olemqsum 9983 exp3val 10011 cvgratz 10980 eucalgval2 11367 ressid2 11607 ressval2 11608 |
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