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Mirrors > Home > MPE Home > Th. List > breqtrid | Structured version Visualization version GIF version |
Description: A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.) |
Ref | Expression |
---|---|
breqtrid.1 | ⊢ 𝐴𝑅𝐵 |
breqtrid.2 | ⊢ (𝜑 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
breqtrid | ⊢ (𝜑 → 𝐴𝑅𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breqtrid.1 | . . 3 ⊢ 𝐴𝑅𝐵 | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → 𝐴𝑅𝐵) |
3 | breqtrid.2 | . 2 ⊢ (𝜑 → 𝐵 = 𝐶) | |
4 | 2, 3 | breqtrd 5100 | 1 ⊢ (𝜑 → 𝐴𝑅𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 class class class wbr 5074 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5075 |
This theorem is referenced by: breqtrrid 5112 phplem3OLD 9002 xlemul1a 13022 phicl2 16469 sinq12ge0 25665 siilem1 29213 nmbdfnlbi 30411 nmcfnlbi 30414 unierri 30466 leoprf2 30489 leoprf 30490 ballotlemic 32473 ballotlem1c 32474 sumnnodd 43171 |
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