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Mirrors > Home > MPE Home > Th. List > breq12 | Structured version Visualization version GIF version |
Description: Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.) |
Ref | Expression |
---|---|
breq12 | ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 4586 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶)) | |
2 | breq2 4587 | . 2 ⊢ (𝐶 = 𝐷 → (𝐵𝑅𝐶 ↔ 𝐵𝑅𝐷)) | |
3 | 1, 2 | sylan9bb 732 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 class class class wbr 4583 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 |
This theorem is referenced by: breq12i 4592 breq12d 4596 breqan12d 4599 rbropapd 4939 posn 5110 dfrel4 5504 isopolem 6495 poxp 7176 soxp 7177 fnse 7181 ecopover 7738 ecopoverOLD 7739 canth2g 7999 infxpen 8720 sornom 8982 dcomex 9152 zorn2lem6 9206 brdom6disj 9235 fpwwe2 9344 rankcf 9478 ltresr 9840 ltxrlt 9987 wloglei 10439 ltxr 11825 xrltnr 11829 xrltnsym 11846 xrlttri 11848 xrlttr 11849 brfi1uzind 13135 brfi1indALT 13137 brfi1uzindOLD 13141 brfi1indALTOLD 13143 f1olecpbl 16010 isfull 16393 isfth 16397 prslem 16754 pslem 17029 dirtr 17059 xrsdsval 19609 dvcvx 23587 axcontlem9 25652 iscusgra 25985 sizeusglecusg 26014 iswlkon 26062 istrlon 26071 ispth 26098 isspth 26099 ispthon 26106 0pthonv 26111 isspthon 26113 1pthon2v 26123 2pthon3v 26134 usg2wlk 26145 usg2wlkon 26146 constr3cyclpe 26191 3v3e3cycl2 26192 isclwlk0 26282 isrusgra 26454 iseupa 26492 2sqmo 28980 mclsppslem 30734 dfpo2 30898 fununiq 30913 elfix2 31181 poimirlem10 32589 poimirlem11 32590 monotoddzzfi 36525 isrusgr 40761 1wlk2f 40834 istrlson 40914 upgrwlkdvspth 40945 ispthson 40948 isspthson 40949 crctcsh1wlk 41025 crctcsh 41027 2pthon3v-av 41150 umgr2wlk 41156 0pthonv-av 41297 1pthon2v-av 41320 uhgr3cyclex 41349 lindepsnlininds 42035 |
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