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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 5151 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 𝐶𝑅𝐷) | 
| 5 | 1, 4 | sylib 218 | 1 ⊢ (𝜑 → 𝐶𝑅𝐷) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 class class class wbr 5142 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 | 
| This theorem is referenced by: eqbrtrrid 5178 breqtrdi 5183 ssenen 9192 adderpq 10997 mulerpq 10998 ltaddnq 11015 ege2le3 16127 ovolfiniun 25537 dvfsumlem3 26070 basellem9 27133 pnt2 27658 pnt 27659 siilem1 30871 omndaddr 33085 ogrpaddltrd 33097 sn-0ne2 42441 | 
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