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Mirrors > Home > ILE Home > Th. List > ifbid | GIF version |
Description: Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.) |
Ref | Expression |
---|---|
ifbid.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
ifbid | ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐴, 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifbid.1 | . 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | ifbi 3569 | . 2 ⊢ ((𝜓 ↔ 𝜒) → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐴, 𝐵)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐴, 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-11 1517 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-if 3550 |
This theorem is referenced by: ifbieq1d 3571 ifbieq2d 3573 ifbieq12d 3575 ifandc 3587 ifordc 3588 pw2f1odclem 6857 nnnninf 7149 nnnninf2 7150 nnnninfeq 7151 nninfisollemne 7154 nninfisol 7156 fodjum 7169 fodju0 7170 fodjuomni 7172 fodjumkv 7183 nninfwlporlemd 7195 nninfwlpor 7197 nninfwlpoimlemg 7198 nninfwlpoimlemginf 7199 nninfwlpoim 7201 xaddval 9870 0tonninf 10465 1tonninf 10466 sumeq1 11390 summodc 11418 zsumdc 11419 fsum3 11422 isumss 11426 sumsplitdc 11467 prodeq1f 11587 zproddc 11614 fprodseq 11618 pcmpt 12370 pcmpt2 12371 pcfac 12377 lgsval 14842 lgsneg 14862 lgsdilem 14865 lgsdir2 14871 lgsdir 14873 bj-charfunbi 15000 subctctexmid 15188 nninfalllem1 15195 nninfsellemdc 15197 nninfself 15200 nninfsellemeq 15201 nninfsellemqall 15202 nninfsellemeqinf 15203 nninfomni 15206 nninffeq 15207 dceqnconst 15246 dcapnconst 15247 |
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