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Mirrors > Home > MPE Home > Th. List > 3brtr3g | Structured version Visualization version 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 5059 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 𝐶𝑅𝐷) |
5 | 1, 4 | sylib 221 | 1 ⊢ (𝜑 → 𝐶𝑅𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 class class class wbr 5050 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3067 df-v 3407 df-dif 3866 df-un 3868 df-nul 4235 df-if 4437 df-sn 4539 df-pr 4541 df-op 4545 df-br 5051 |
This theorem is referenced by: eqbrtrrid 5086 breqtrdi 5091 ssenen 8817 adderpq 10567 mulerpq 10568 ltaddnq 10585 ege2le3 15648 ovolfiniun 24395 dvfsumlem3 24922 basellem9 25968 pnt2 26491 pnt 26492 siilem1 28929 omndaddr 31049 ogrpaddltrd 31061 sn-0ne2 40095 |
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