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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 4592 | . 2 ⊢ (𝐶𝑅𝐷 ↔ 𝐴𝑅𝐵) |
5 | 1, 4 | sylibr 223 | 1 ⊢ (𝜑 → 𝐶𝑅𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 class class class wbr 4583 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 |
This theorem is referenced by: syl5eqbr 4618 limensuci 8021 infensuc 8023 rlimneg 14225 isumsup2 14417 crth 15321 4sqlem6 15485 gzrngunit 19631 matgsum 20062 ovolunlem1a 23071 ovolfiniun 23076 ioombl1lem1 23133 ioombl1lem4 23136 iblss 23377 itgle 23382 dvfsumlem3 23595 emcllem6 24527 gausslemma2dlem0f 24886 gausslemma2dlem0g 24887 pntpbnd1a 25074 ostth2lem4 25125 omsmon 29687 itg2gt0cn 32635 dalem-cly 33975 dalem10 33977 fourierdlem103 39102 fourierdlem104 39103 |
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