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Mirrors > Home > MPE Home > Th. List > 3brtr4g | 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 |
---|---|
3brtr4g.1 | ⊢ (𝜑 → 𝐴𝑅𝐵) |
3brtr4g.2 | ⊢ 𝐶 = 𝐴 |
3brtr4g.3 | ⊢ 𝐷 = 𝐵 |
Ref | Expression |
---|---|
3brtr4g | ⊢ (𝜑 → 𝐶𝑅𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3brtr4g.1 | . 2 ⊢ (𝜑 → 𝐴𝑅𝐵) | |
2 | 3brtr4g.2 | . . 3 ⊢ 𝐶 = 𝐴 | |
3 | 3brtr4g.3 | . . 3 ⊢ 𝐷 = 𝐵 | |
4 | 2, 3 | breq12i 5059 | . 2 ⊢ (𝐶𝑅𝐷 ↔ 𝐴𝑅𝐵) |
5 | 1, 4 | sylibr 237 | 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: eqbrtrid 5085 enrefnn 8721 enpr2d 8722 limensuci 8819 infensuc 8821 djuen 9780 djudom1 9793 rlimneg 15207 isumsup2 15407 crth 16328 4sqlem6 16493 gzrngunit 20426 matgsum 21331 ovolunlem1a 24390 ovolfiniun 24395 ioombl1lem1 24452 ioombl1lem4 24455 iblss 24699 itgle 24704 dvfsumlem3 24922 emcllem6 25880 gausslemma2dlem0f 26239 gausslemma2dlem0g 26240 pntpbnd1a 26463 ostth2lem4 26514 omsmon 31974 noinfbnd2lem1 33667 itg2gt0cn 35567 dalem-cly 37420 dalem10 37422 fourierdlem103 43423 fourierdlem104 43424 |
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