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Mirrors > Home > MPE Home > Th. List > 2ralbii | Structured version Visualization version GIF version |
Description: Inference adding two restricted universal quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.) |
Ref | Expression |
---|---|
2ralbii.1 | ⊢ (𝜑 ↔ 𝜓) |
Ref | Expression |
---|---|
2ralbii | ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2ralbii.1 | . . 3 ⊢ (𝜑 ↔ 𝜓) | |
2 | 1 | ralbii 3094 | . 2 ⊢ (∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓) |
3 | 2 | ralbii 3094 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∀wral 3062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 |
This theorem depends on definitions: df-bi 206 df-ral 3063 |
This theorem is referenced by: 3ralbii 3131 ralnex3 3135 rmo4f 3732 2reu4lem 4526 cnvso 6288 fununi 6624 dff14a 7269 isocnv2 7328 f1opr 7465 sorpss 7718 xpord3inddlem 8140 tpossym 8243 dford2 9615 isffth2 17867 ispos2 18268 issubm 18684 cntzrec 19200 oppgsubm 19229 opprirred 20236 opprsubrg 20340 isdomn5 20917 gsummatr01lem3 22159 gsummatr01 22161 isbasis2g 22451 ist0-3 22849 isfbas2 23339 isclmp 24613 addsproplem4 27456 addsproplem6 27458 addsprop 27460 negsproplem4 27505 negsproplem6 27507 negsprop 27509 mulsprop 27586 dfadj2 31138 adjval2 31144 cnlnadjeui 31330 adjbdln 31336 isarchi 32328 prmidl0 32569 dff15 34087 iccllysconn 34241 dfso3 34689 elpotr 34753 dfon2 34764 idinxpss 37181 inxpssidinxp 37185 idinxpssinxp 37186 dfeldisj5 37591 isltrn2N 38991 fphpd 41554 isdomn3 41946 fiinfi 42324 ntrk1k3eqk13 42801 ordelordALT 43298 disjinfi 43891 issubmgm 46559 opprsubrng 46738 rhmimasubrng 46745 cntzsubrng 46746 isthinc2 47642 isthinc3 47643 |
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