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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 3093 | . 2 ⊢ (∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓) |
3 | 2 | ralbii 3093 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∀wral 3061 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 |
This theorem depends on definitions: df-bi 206 df-ral 3062 |
This theorem is referenced by: 3ralbii 3130 ralnex3 3134 rmo4f 3731 2reu4lem 4525 cnvso 6287 fununi 6623 dff14a 7268 isocnv2 7327 f1opr 7464 sorpss 7717 xpord3inddlem 8139 tpossym 8242 dford2 9614 isffth2 17866 ispos2 18267 issubm 18683 cntzrec 19199 oppgsubm 19228 opprirred 20235 opprsubrg 20339 isdomn5 20916 gsummatr01lem3 22158 gsummatr01 22160 isbasis2g 22450 ist0-3 22848 isfbas2 23338 isclmp 24612 addsproplem4 27453 addsproplem6 27455 addsprop 27457 negsproplem4 27502 negsproplem6 27504 negsprop 27506 mulsprop 27583 dfadj2 31133 adjval2 31139 cnlnadjeui 31325 adjbdln 31331 isarchi 32323 prmidl0 32564 dff15 34082 iccllysconn 34236 dfso3 34684 elpotr 34748 dfon2 34759 idinxpss 37176 inxpssidinxp 37180 idinxpssinxp 37181 dfeldisj5 37586 isltrn2N 38986 fphpd 41544 isdomn3 41936 fiinfi 42314 ntrk1k3eqk13 42791 ordelordALT 43288 disjinfi 43881 issubmgm 46549 opprsubrng 46728 rhmimasubrng 46735 cntzsubrng 46736 isthinc2 47632 isthinc3 47633 |
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