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Mirrors > Home > MPE Home > Th. List > releqd | Structured version Visualization version GIF version |
Description: Equality deduction for the relation predicate. (Contributed by NM, 8-Mar-2014.) |
Ref | Expression |
---|---|
releqd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
releqd | ⊢ (𝜑 → (Rel 𝐴 ↔ Rel 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | releqd.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | releq 5653 | . 2 ⊢ (𝐴 = 𝐵 → (Rel 𝐴 ↔ Rel 𝐵)) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (Rel 𝐴 ↔ Rel 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 Rel wrel 5562 |
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-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-in 3945 df-ss 3954 df-rel 5564 |
This theorem is referenced by: dftpos3 7912 tposfo2 7917 tposf12 7919 relexp0rel 14398 relexprelg 14399 relexpaddg 14414 imasaddfnlem 16803 imasvscafn 16812 cicer 17078 joindmss 17619 meetdmss 17633 mattpostpos 21065 cnextrel 22673 perpln1 26498 perpln2 26499 relfae 31508 satfrel 32616 dibvalrel 38301 dicvalrelN 38323 diclspsn 38332 dihvalrel 38417 dih1 38424 dihmeetlem4preN 38444 |
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