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Mirrors > Home > MPE Home > Th. List > breq12i | Structured version Visualization version GIF version |
Description: Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Eric Schmidt, 4-Apr-2007.) |
Ref | Expression |
---|---|
breq1i.1 | ⊢ 𝐴 = 𝐵 |
breq12i.2 | ⊢ 𝐶 = 𝐷 |
Ref | Expression |
---|---|
breq12i | ⊢ (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | breq12i.2 | . 2 ⊢ 𝐶 = 𝐷 | |
3 | breq12 5153 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)) | |
4 | 1, 2, 3 | mp2an 692 | 1 ⊢ (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = 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: 3brtr3g 5181 3brtr4g 5182 caovord2 7645 domunfican 9359 ltsonq 11007 ltanq 11009 ltmnq 11010 prlem934 11071 prlem936 11085 ltsosr 11132 ltasr 11138 ltneg 11761 leneg 11764 inelr 12254 lt2sqi 14225 le2sqi 14226 nn0le2msqi 14303 2sqreuop 27521 2sqreuopnn 27522 2sqreuoplt 27523 2sqreuopltb 27524 2sqreuopnnlt 27525 2sqreuopnnltb 27526 axlowdimlem6 28977 upgrwlkcompim 29676 clwlkcompbp 29815 mdsldmd1i 32360 divcnvlin 35713 ditgeq123i 36191 cbvditgvw2 36232 relowlpssretop 37347 2ap1caineq 42127 fsumlessf 45533 climlimsupcex 45725 liminfltlimsupex 45737 liminflelimsupcex 45753 sge0xaddlem2 46390 eubrdm 46986 isgrlim2 47886 iscmgmALT 48068 iscsgrpALT 48070 |
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