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Mirrors > Home > MPE Home > Th. List > breqtrrid | Structured version Visualization version GIF version |
Description: A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.) |
Ref | Expression |
---|---|
breqtrrid.1 | ⊢ 𝐴𝑅𝐵 |
breqtrrid.2 | ⊢ (𝜑 → 𝐶 = 𝐵) |
Ref | Expression |
---|---|
breqtrrid | ⊢ (𝜑 → 𝐴𝑅𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breqtrrid.1 | . 2 ⊢ 𝐴𝑅𝐵 | |
2 | breqtrrid.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐵) | |
3 | 2 | eqcomd 2741 | . 2 ⊢ (𝜑 → 𝐵 = 𝐶) |
4 | 1, 3 | breqtrid 5185 | 1 ⊢ (𝜑 → 𝐴𝑅𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 class class class wbr 5148 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 |
This theorem is referenced by: r1sdom 9812 alephordilem1 10111 mulge0 11779 xsubge0 13300 xmulgt0 13322 xmulge0 13323 xlemul1a 13327 sqlecan 14245 bernneq 14265 hashge1 14425 hashge2el2dif 14516 cnpart 15276 sqrt0 15277 bitsfzo 16469 bitsmod 16470 bitsinv1lem 16475 pcge0 16896 prmreclem4 16953 prmreclem5 16954 isnzr2hash 20536 isabvd 20830 abvtrivd 20850 nmolb2d 24755 nmoi 24765 nmoleub 24768 nmo0 24772 ovolge0 25530 itg1ge0a 25761 fta1g 26224 plyrem 26362 taylfval 26415 abelthlem2 26491 sinq12ge0 26565 relogrn 26618 logneg 26645 cxpge0 26740 amgmlem 27048 bposlem5 27347 lgsdir2lem2 27385 2lgsoddprmlem3 27473 rpvmasumlem 27546 mulsge0d 28187 expsgt0 28430 eupth2lem3lem3 30259 eupth2lemb 30266 blocnilem 30833 pjssge0ii 31711 unierri 32133 xlt2addrd 32769 2sqr3minply 33753 locfinref 33802 esumcst 34044 ballotlem5 34481 poimirlem23 37630 poimirlem25 37632 poimirlem26 37633 poimirlem27 37634 poimirlem28 37635 itgaddnclem2 37666 pell14qrgt0 42847 monotoddzzfi 42931 rmxypos 42936 rmygeid 42953 stoweidlem18 45974 stoweidlem55 46011 wallispi2lem1 46027 fourierdlem62 46124 fourierdlem103 46165 fourierdlem104 46166 fourierswlem 46186 2ltceilhalf 47950 pgrpgt2nabl 48211 pw2m1lepw2m1 48366 amgmwlem 49033 |
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