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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 5065 | . 2 ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)) |
4 | breq123d.2 | . . 3 ⊢ (𝜑 → 𝑅 = 𝑆) | |
5 | 4 | breqd 5063 | . 2 ⊢ (𝜑 → (𝐵𝑅𝐷 ↔ 𝐵𝑆𝐷)) |
6 | 3, 5 | bitrd 281 | 1 ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑆𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 class class class wbr 5052 |
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 2793 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3488 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-sn 4554 df-pr 4556 df-op 4560 df-br 5053 |
This theorem is referenced by: sbcbr123 5106 fmptco 6877 xpsle 16835 invfuc 17227 yonedainv 17514 opphllem3 26521 lmif 26557 islmib 26559 iscgra 26581 isinag 26610 fmptcof2 30388 submomnd 30718 sgnsv 30809 inftmrel 30816 isinftm 30817 submarchi 30822 suborng 30895 uncov 34907 iscvlat 36491 paddfval 36965 lhpset 37163 tendofset 37926 diaffval 38198 fnwe2val 39741 aomclem8 39753 afv2eq12d 43504 |
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