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Mirrors > Home > MPE Home > Th. List > rexralbidv | Structured version Visualization version GIF version |
Description: Formula-building rule for restricted quantifiers (deduction form). (Contributed by NM, 28-Jan-2006.) |
Ref | Expression |
---|---|
2ralbidv.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
rexralbidv | ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2ralbidv.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | 1 | ralbidv 3176 | . 2 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 𝜓 ↔ ∀𝑦 ∈ 𝐵 𝜒)) |
3 | 2 | rexbidv 3177 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∀wral 3059 ∃wrex 3068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-ral 3060 df-rex 3069 |
This theorem is referenced by: freq1 5656 rexfiuz 15383 cau3lem 15390 caubnd2 15393 climi 15543 rlimi 15546 o1lo1 15570 2clim 15605 lo1le 15685 caucvgrlem 15706 caurcvgr 15707 caucvgb 15713 vdwlem10 17024 vdwlem13 17027 pmatcollpw2lem 22799 neiptopnei 23156 lmcvg 23286 lmss 23322 elpt 23596 elptr 23597 txlm 23672 tsmsi 24158 ustuqtop4 24269 isucn 24303 isucn2 24304 ucnima 24306 metcnpi 24573 metcnpi2 24574 metucn 24600 xrge0tsms 24870 elcncf 24929 cncfi 24934 lmmcvg 25309 lhop1 26068 ulmval 26438 ulmi 26444 ulmcaulem 26452 ulmdvlem3 26460 pntibnd 27652 pntlem3 27668 pntleml 27670 axtgcont1 28491 perpln1 28733 perpln2 28734 isperp 28735 brbtwn 28929 uvtx01vtx 29429 isgrpo 30526 ubthlem3 30901 ubth 30902 hcau 31213 hcaucvg 31215 hlimi 31217 hlimconvi 31220 hlim2 31221 elcnop 31886 elcnfn 31911 cnopc 31942 cnfnc 31959 lnopcon 32064 lnfncon 32085 riesz1 32094 xrge0tsmsd 33048 signsply0 34545 unblimceq0 36490 cvgcau 45441 limcleqr 45600 addlimc 45604 0ellimcdiv 45605 climd 45628 climisp 45702 lmbr3 45703 climrescn 45704 climxrrelem 45705 climxrre 45706 xlimpnfxnegmnf 45770 xlimxrre 45787 xlimmnf 45797 xlimpnf 45798 xlimmnfmpt 45799 xlimpnfmpt 45800 dfxlim2 45804 cncfshift 45830 cncfperiod 45835 ioodvbdlimc1lem1 45887 ioodvbdlimc1lem2 45888 ioodvbdlimc2lem 45890 fourierdlem68 46130 fourierdlem87 46149 fourierdlem103 46165 fourierdlem104 46166 etransclem48 46238 |
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