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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 5156 | . 2 ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)) |
| 4 | breq123d.2 | . . 3 ⊢ (𝜑 → 𝑅 = 𝑆) | |
| 5 | 4 | breqd 5154 | . 2 ⊢ (𝜑 → (𝐵𝑅𝐷 ↔ 𝐵𝑆𝐷)) |
| 6 | 3, 5 | bitrd 279 | 1 ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑆𝐷)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 class class class wbr 5143 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 |
| This theorem is referenced by: sbcbr123 5197 fmptco 7149 xpsle 17624 invfuc 18022 yonedainv 18326 opphllem3 28757 lmif 28793 islmib 28795 iscgra 28817 isinag 28846 fmptcof2 32667 submomnd 33087 sgnsv 33180 inftmrel 33187 isinftm 33188 submarchi 33193 rlocval 33263 suborng 33345 rprmval 33544 weiunlem1 36463 uncov 37608 iscvlat 39324 paddfval 39799 lhpset 39997 tendofset 40760 diaffval 41032 fnwe2val 43061 aomclem8 43073 afv2eq12d 47227 |
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