|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 5152 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 𝐶𝑅𝐷) | 
| 5 | 1, 4 | sylib 218 | 1 ⊢ (𝜑 → 𝐶𝑅𝐷) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 class class class wbr 5143 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 | 
| This theorem is referenced by: eqbrtrrid 5179 breqtrdi 5184 ssenen 9191 adderpq 10996 mulerpq 10997 ltaddnq 11014 ege2le3 16126 ovolfiniun 25536 dvfsumlem3 26069 basellem9 27132 pnt2 27657 pnt 27658 siilem1 30870 omndaddr 33084 ogrpaddltrd 33096 sn-0ne2 42436 | 
| Copyright terms: Public domain | W3C validator |