![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > eqbrtrrd | GIF version |
Description: Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.) |
Ref | Expression |
---|---|
eqbrtrrd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
eqbrtrrd.2 | ⊢ (𝜑 → 𝐴𝑅𝐶) |
Ref | Expression |
---|---|
eqbrtrrd | ⊢ (𝜑 → 𝐵𝑅𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqbrtrrd.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | 1 | eqcomd 2088 | . 2 ⊢ (𝜑 → 𝐵 = 𝐴) |
3 | eqbrtrrd.2 | . 2 ⊢ (𝜑 → 𝐴𝑅𝐶) | |
4 | 2, 3 | eqbrtrd 3831 | 1 ⊢ (𝜑 → 𝐵𝑅𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1285 class class class wbr 3811 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-v 2614 df-un 2988 df-sn 3428 df-pr 3429 df-op 3431 df-br 3812 |
This theorem is referenced by: dftpos4 5959 phpm 6510 unsnfidcex 6556 fisseneq 6566 f1finf1o 6579 prmuloclemcalc 7026 mullocprlem 7031 cauappcvgprlemladdfl 7116 caucvgprlemopl 7130 caucvgprprlemloccalc 7145 caucvgprprlemopl 7158 ltadd1sr 7224 axarch 7328 lemulge11 8220 modqmuladdim 9662 ltexp2a 9843 leexp2a 9844 nnlesq 9893 faclbnd6 9986 facavg 9988 fiprsshashgt1 10059 cvg1nlemcxze 10241 resqrexlemover 10269 resqrexlemlo 10272 resqrexlemnmsq 10276 resqrexlemnm 10277 leabs 10333 abs3dif 10364 abs2dif 10365 maxabslemlub 10466 maxltsup 10477 recn2 10528 imcn2 10529 iiserex 10550 divalglemnqt 10699 mulgcd 10784 dvdssqlem 10798 nn0seqcvgd 10802 mulgcddvds 10855 rpdvds 10860 pw2dvdseulemle 10924 sqrt2irraplemnn 10936 qden1elz 10962 phimullem 10980 hashgcdlem 10982 hashgcdeq 10983 |
Copyright terms: Public domain | W3C validator |