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Mirrors > Home > MPE Home > Th. List > breq123d | Structured version Visualization version GIF version |
Description: Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.) |
Ref | Expression |
---|---|
breq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
breq123d.2 | ⊢ (𝜑 → 𝑅 = 𝑆) |
breq123d.3 | ⊢ (𝜑 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
breq123d | ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑆𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | breq123d.3 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
3 | 1, 2 | breq12d 5043 | . 2 ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)) |
4 | breq123d.2 | . . 3 ⊢ (𝜑 → 𝑅 = 𝑆) | |
5 | 4 | breqd 5041 | . 2 ⊢ (𝜑 → (𝐵𝑅𝐷 ↔ 𝐵𝑆𝐷)) |
6 | 3, 5 | bitrd 282 | 1 ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑆𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 class class class wbr 5030 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-un 3886 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 |
This theorem is referenced by: sbcbr123 5084 fmptco 6868 xpsle 16844 invfuc 17236 yonedainv 17523 opphllem3 26543 lmif 26579 islmib 26581 iscgra 26603 isinag 26632 fmptcof2 30420 submomnd 30761 sgnsv 30852 inftmrel 30859 isinftm 30860 submarchi 30865 suborng 30939 rprmval 31072 uncov 35038 iscvlat 36619 paddfval 37093 lhpset 37291 tendofset 38054 diaffval 38326 fnwe2val 39993 aomclem8 40005 afv2eq12d 43771 |
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