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Mirrors > Home > NFE Home > Th. List > breq2 | GIF version |
Description: Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.) |
Ref | Expression |
---|---|
breq2 | ⊢ (A = B → (CRA ↔ CRB)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq2 4579 | . . 3 ⊢ (A = B → 〈C, A〉 = 〈C, B〉) | |
2 | 1 | eleq1d 2419 | . 2 ⊢ (A = B → (〈C, A〉 ∈ R ↔ 〈C, B〉 ∈ R)) |
3 | df-br 4640 | . 2 ⊢ (CRA ↔ 〈C, A〉 ∈ R) | |
4 | df-br 4640 | . 2 ⊢ (CRB ↔ 〈C, B〉 ∈ R) | |
5 | 2, 3, 4 | 3bitr4g 279 | 1 ⊢ (A = B → (CRA ↔ CRB)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 = wceq 1642 ∈ wcel 1710 〈cop 4561 class class class wbr 4639 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-addc 4378 df-nnc 4379 df-phi 4565 df-op 4566 df-br 4640 |
This theorem is referenced by: breq12 4644 breq2i 4647 breq2d 4651 nbrne1 4656 elima 4754 vtoclr 4816 brco 4883 brcnv 4892 dfdmf 4905 dfrnf 4962 dfres2 5002 elimasn 5019 intirr 5029 df2nd2 5111 dffun3 5120 dffun6f 5123 fun11 5159 fv3 5341 tz6.12-1 5344 tz6.12c 5347 tz6.12i 5348 funfv2f 5377 isorel 5489 isocnv 5491 isotr 5495 f1oiso 5499 f1oiso2 5500 opbr1st 5501 opbr2nd 5502 caovord 5629 brsnsi 5773 brsnsi2 5776 trtxp 5781 addcfnex 5824 brfns 5833 qrpprod 5836 fnfullfunlem1 5856 fvfullfunlem1 5861 clos1conn 5879 clos1basesuc 5882 trd 5921 extd 5923 symd 5924 antid 5929 connexd 5931 idssen 6040 enadj 6060 enpw1lem1 6061 enpw1 6062 enmap2lem1 6063 enmap2 6068 enmap1lem1 6069 enprmaplem3 6078 enpw 6087 ovmuc 6130 mucex 6133 1cnc 6139 ovcelem1 6171 ceex 6174 sbthlem2 6204 dflec2 6210 lectr 6211 nc0le1 6216 leconnnc 6218 taddc 6229 tlecg 6230 ce2le 6233 nclenn 6249 nnltp1clem1 6261 lecadd2 6266 ncslesuc 6267 nmembers1lem1 6268 nmembers1lem3 6270 nmembers1 6271 nchoicelem4 6292 nchoicelem11 6299 nchoicelem19 6307 fnfrec 6320 |
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