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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 4588 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)) | |
4 | 1, 2, 3 | mp2an 704 | 1 ⊢ (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = 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: 3brtr3g 4616 3brtr4g 4617 caovord2 6744 domunfican 8118 ltsonq 9670 ltanq 9672 ltmnq 9673 prlem934 9734 prlem936 9748 ltsosr 9794 ltasr 9800 ltneg 10407 leneg 10410 inelr 10887 lt2sqi 12814 le2sqi 12815 nn0le2msqi 12916 axlowdimlem6 25627 mdsldmd1i 28574 divcnvlin 30871 relowlpssretop 32388 fsumlessf 38644 sge0xaddlem2 39327 upgr1wlkcompim 40851 iscmgmALT 41650 iscsgrpALT 41652 |
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