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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 5071 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶)) | |
2 | breq2 5072 | . 2 ⊢ (𝐶 = 𝐷 → (𝐵𝑅𝐶 ↔ 𝐵𝑅𝐷)) | |
3 | 1, 2 | sylan9bb 512 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 class class class wbr 5068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 |
This theorem is referenced by: breq12i 5077 breq12d 5081 breqan12d 5084 rbropapd 5451 posn 5639 dfrel4 6050 isopolem 7100 poxp 7824 soxp 7825 fnse 7829 ecopover 8403 canth2g 8673 infxpen 9442 sornom 9701 dcomex 9871 zorn2lem6 9925 brdom6disj 9956 fpwwe2 10067 rankcf 10201 ltresr 10564 ltxrlt 10713 wloglei 11174 ltxr 12513 xrltnr 12517 xrltnsym 12533 xrlttri 12535 xrlttr 12536 brfi1uzind 13859 brfi1indALT 13861 f1olecpbl 16802 isfull 17182 isfth 17186 prslem 17543 pslem 17818 dirtr 17848 xrsdsval 20591 dvcvx 24619 2sqmo 26015 2sqreultblem 26026 2sqreunnltblem 26029 2sqreuopb 26046 axcontlem9 26760 isrusgr 27345 wlk2f 27413 istrlson 27490 upgrwlkdvspth 27522 ispthson 27525 isspthson 27526 crctcshwlk 27602 crctcsh 27604 2pthon3v 27724 umgr2wlk 27730 0pthonv 27910 1pthon2v 27934 uhgr3cyclex 27963 brfinext 31045 mclsppslem 32832 dfpo2 32993 fununiq 33014 slerec 33279 elfix2 33367 poimirlem10 34904 poimirlem11 34905 factwoffsmonot 39105 monotoddzzfi 39546 or2expropbi 43276 dfatcolem 43461 sprsymrelfolem2 43662 poprelb 43693 lindepsnlininds 44514 |
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