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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 5157 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 𝐶𝑅𝐷) |
5 | 1, 4 | sylib 218 | 1 ⊢ (𝜑 → 𝐶𝑅𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 class class class wbr 5148 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 |
This theorem is referenced by: eqbrtrrid 5184 breqtrdi 5189 ssenen 9190 adderpq 10994 mulerpq 10995 ltaddnq 11012 ege2le3 16123 ovolfiniun 25550 dvfsumlem3 26084 basellem9 27147 pnt2 27672 pnt 27673 siilem1 30880 omndaddr 33067 ogrpaddltrd 33079 sn-0ne2 42413 |
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