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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 5077 | . 2 ⊢ (𝐶𝑅𝐷 ↔ 𝐴𝑅𝐵) |
5 | 1, 4 | sylibr 236 | 1 ⊢ (𝜑 → 𝐶𝑅𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 class class class wbr 5068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 |
This theorem is referenced by: eqbrtrid 5103 enpr2d 8599 limensuci 8695 infensuc 8697 djuen 9597 djudom1 9610 rlimneg 15005 isumsup2 15203 crth 16117 4sqlem6 16281 gzrngunit 20613 matgsum 21048 ovolunlem1a 24099 ovolfiniun 24104 ioombl1lem1 24161 ioombl1lem4 24164 iblss 24407 itgle 24412 dvfsumlem3 24627 emcllem6 25580 gausslemma2dlem0f 25939 gausslemma2dlem0g 25940 pntpbnd1a 26163 ostth2lem4 26214 omsmon 31558 itg2gt0cn 34949 dalem-cly 36809 dalem10 36811 fourierdlem103 42501 fourierdlem104 42502 |
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