Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 5201 | . 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 196 = wceq 1483 Rel wrel 5119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-in 3581 df-ss 3588 df-rel 5121 |
This theorem is referenced by: dftpos3 7370 tposfo2 7375 tposf12 7377 relexp0rel 13777 relexprelg 13778 relexpaddg 13793 imasaddfnlem 16188 imasvscafn 16197 cicer 16466 joindmss 17007 meetdmss 17021 mattpostpos 20260 cnextrel 21867 perpln1 25605 perpln2 25606 relfae 30310 dibvalrel 36452 dicvalrelN 36474 diclspsn 36483 dihvalrel 36568 dih1 36575 dihmeetlem4preN 36595 |
Copyright terms: Public domain | W3C validator |