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Mirrors > Home > ILE Home > Th. List > 3brtr4d | GIF version |
Description: Substitution of equality into both sides of a binary relation. (Contributed by NM, 21-Feb-2005.) |
Ref | Expression |
---|---|
3brtr4d.1 | ⊢ (𝜑 → 𝐴𝑅𝐵) |
3brtr4d.2 | ⊢ (𝜑 → 𝐶 = 𝐴) |
3brtr4d.3 | ⊢ (𝜑 → 𝐷 = 𝐵) |
Ref | Expression |
---|---|
3brtr4d | ⊢ (𝜑 → 𝐶𝑅𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3brtr4d.1 | . 2 ⊢ (𝜑 → 𝐴𝑅𝐵) | |
2 | 3brtr4d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐴) | |
3 | 3brtr4d.3 | . . 3 ⊢ (𝜑 → 𝐷 = 𝐵) | |
4 | 2, 3 | breq12d 3824 | . 2 ⊢ (𝜑 → (𝐶𝑅𝐷 ↔ 𝐴𝑅𝐵)) |
5 | 1, 4 | mpbird 165 | 1 ⊢ (𝜑 → 𝐶𝑅𝐷) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1285 class class class wbr 3811 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-v 2614 df-un 2988 df-sn 3428 df-pr 3429 df-op 3431 df-br 3812 |
This theorem is referenced by: f1oiso2 5545 prarloclemarch2 6881 caucvgprprlemmu 7157 caucvgsrlembound 7242 mulap0 8021 lediv12a 8249 recp1lt1 8254 fldiv4p1lem1div2 9601 intfracq 9616 modqmulnn 9638 addmodlteq 9694 frecfzennn 9722 monoord2 9771 expgt1 9830 leexp2r 9846 leexp1a 9847 bernneq 9909 faclbnd 9984 faclbnd6 9987 facubnd 9988 hashunlem 10047 sqrtgt0 10294 absrele 10343 absimle 10344 abstri 10364 abs2difabs 10368 climsqz 10547 climsqz2 10548 phibnd 10973 |
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