Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > raleq | GIF version |
Description: Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.) |
Ref | Expression |
---|---|
raleq | ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2281 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2281 | . 2 ⊢ Ⅎ𝑥𝐵 | |
3 | 1, 2 | raleqf 2622 | 1 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1331 ∀wral 2416 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 |
This theorem is referenced by: raleqi 2630 raleqdv 2632 raleqbi1dv 2634 sbralie 2670 inteq 3774 iineq1 3827 bnd2 4097 frforeq2 4267 weeq2 4279 ordeq 4294 reg2exmid 4451 reg3exmid 4494 omsinds 4535 fncnv 5189 funimaexglem 5206 isoeq4 5705 acexmidlemv 5772 tfrlem1 6205 tfr0dm 6219 tfrlemisucaccv 6222 tfrlemi1 6229 tfrlemi14d 6230 tfrexlem 6231 tfr1onlemsucaccv 6238 tfr1onlemaccex 6245 tfr1onlemres 6246 tfrcllemsucaccv 6251 tfrcllembxssdm 6253 tfrcllemaccex 6258 tfrcllemres 6259 tfrcldm 6260 ixpeq1 6603 ac6sfi 6792 fimax2gtri 6795 supeq1 6873 supeq2 6876 isomni 7008 ismkv 7027 sup3exmid 8715 rexanuz 10760 rexfiuz 10761 fimaxre2 10998 modfsummod 11227 cnprcl2k 12375 ispsmet 12492 ismet 12513 isxmet 12514 cncfval 12728 dvcn 12833 setindis 13165 bdsetindis 13167 strcoll2 13181 |
Copyright terms: Public domain | W3C validator |