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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 2306 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2306 | . 2 ⊢ Ⅎ𝑥𝐵 | |
3 | 1, 2 | raleqf 2655 | 1 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1342 ∀wral 2442 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 |
This theorem is referenced by: raleqi 2663 raleqdv 2665 raleqbi1dv 2667 sbralie 2705 inteq 3821 iineq1 3874 bnd2 4146 frforeq2 4317 weeq2 4329 ordeq 4344 reg2exmid 4507 reg3exmid 4551 omsinds 4593 fncnv 5248 funimaexglem 5265 isoeq4 5766 acexmidlemv 5834 tfrlem1 6267 tfr0dm 6281 tfrlemisucaccv 6284 tfrlemi1 6291 tfrlemi14d 6292 tfrexlem 6293 tfr1onlemsucaccv 6300 tfr1onlemaccex 6307 tfr1onlemres 6308 tfrcllemsucaccv 6313 tfrcllembxssdm 6315 tfrcllemaccex 6320 tfrcllemres 6321 tfrcldm 6322 ixpeq1 6666 ac6sfi 6855 fimax2gtri 6858 dcfi 6937 supeq1 6942 supeq2 6945 nnnninfeq2 7084 isomni 7091 ismkv 7108 iswomni 7120 sup3exmid 8843 rexanuz 10916 rexfiuz 10917 fimaxre2 11154 modfsummod 11385 cnprcl2k 12753 ispsmet 12870 ismet 12891 isxmet 12892 cncfval 13106 dvcn 13211 setindis 13690 bdsetindis 13692 strcoll2 13706 strcollnfALT 13709 |
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