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Mirrors > Home > ILE Home > Th. List > raleqdv | GIF version |
Description: Equality deduction for restricted universal quantifier. (Contributed by NM, 13-Nov-2005.) |
Ref | Expression |
---|---|
raleq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
raleqdv | ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | raleq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | raleq 2603 | . 2 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1316 ∀wral 2393 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 |
This theorem is referenced by: raleqbidv 2615 raleqbidva 2617 omsinds 4505 cbvfo 5654 isoselem 5689 ofrfval 5958 issmo2 6154 smoeq 6155 tfrlemisucaccv 6190 tfr1onlemsucaccv 6206 tfrcllemsucaccv 6219 fzrevral2 9854 fzrevral3 9855 fzshftral 9856 fzoshftral 9983 uzsinds 10183 iseqf1olemqk 10235 seq3f1olemstep 10242 seq3f1olemp 10243 caucvgre 10721 cvg1nlemres 10725 rexuz3 10730 resqrexlemoverl 10761 resqrexlemsqa 10764 resqrexlemex 10765 climconst 11027 climshftlemg 11039 serf0 11089 summodclem2 11119 summodc 11120 zsumdc 11121 mertenslemi1 11272 zsupcllemstep 11565 zsupcllemex 11566 infssuzex 11569 prmind2 11728 ennnfoneleminc 11851 ennnfonelemex 11854 ennnfonelemnn0 11862 ennnfonelemr 11863 lmfval 12288 lmconst 12312 cncnp 12326 metss 12590 sin0pilem2 12790 nninfsellemdc 13133 nninfself 13136 nninfsellemeqinf 13139 nninfomni 13142 |
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