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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 3091 | . 2 ⊢ (∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓) |
3 | 2 | ralbii 3091 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∀wral 3059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 |
This theorem depends on definitions: df-bi 206 df-ral 3060 |
This theorem is referenced by: 3ralbii 3128 ralnex3 3132 rmo4f 3730 2reu4lem 4524 cnvso 6286 fununi 6622 dff14a 7271 isocnv2 7330 f1opr 7467 sorpss 7720 xpord3inddlem 8142 tpossym 8245 dford2 9617 isffth2 17871 ispos2 18272 issubmgm 18627 issubm 18720 cntzrec 19241 oppgsubm 19270 opprirred 20313 opprsubrng 20447 rhmimasubrng 20454 cntzsubrng 20455 opprsubrg 20483 isdomn5 21117 gsummatr01lem3 22379 gsummatr01 22381 isbasis2g 22671 ist0-3 23069 isfbas2 23559 isclmp 24844 addsproplem4 27694 addsproplem6 27696 addsprop 27698 negsproplem4 27744 negsproplem6 27746 negsprop 27748 mulsprop 27825 dfadj2 31405 adjval2 31411 cnlnadjeui 31597 adjbdln 31603 isarchi 32598 prmidl0 32843 dff15 34385 iccllysconn 34539 dfso3 34993 elpotr 35057 dfon2 35068 idinxpss 37484 inxpssidinxp 37488 idinxpssinxp 37489 dfeldisj5 37894 isltrn2N 39294 fphpd 41856 isdomn3 42248 fiinfi 42626 ntrk1k3eqk13 43103 ordelordALT 43600 disjinfi 44189 isthinc2 47729 isthinc3 47730 |
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