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Mirrors > Home > ILE Home > Th. List > 3brtr3g | GIF version |
Description: Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.) |
Ref | Expression |
---|---|
3brtr3g.1 | ⊢ (𝜑 → 𝐴𝑅𝐵) |
3brtr3g.2 | ⊢ 𝐴 = 𝐶 |
3brtr3g.3 | ⊢ 𝐵 = 𝐷 |
Ref | Expression |
---|---|
3brtr3g | ⊢ (𝜑 → 𝐶𝑅𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3brtr3g.1 | . 2 ⊢ (𝜑 → 𝐴𝑅𝐵) | |
2 | 3brtr3g.2 | . . 3 ⊢ 𝐴 = 𝐶 | |
3 | 3brtr3g.3 | . . 3 ⊢ 𝐵 = 𝐷 | |
4 | 2, 3 | breq12i 3998 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 𝐶𝑅𝐷) |
5 | 1, 4 | sylib 121 | 1 ⊢ (𝜑 → 𝐶𝑅𝐷) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 class class class wbr 3989 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-un 3125 df-sn 3589 df-pr 3590 df-op 3592 df-br 3990 |
This theorem is referenced by: eqbrtrrid 4025 breqtrdi 4030 ssenen 6829 endjusym 7073 djuen 7188 ege2le3 11634 |
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