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Mirrors > Home > MPE Home > Th. List > r19.41v | Structured version Visualization version GIF version |
Description: Restricted quantifier version 19.41v 1954. Version of r19.41 3261 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 17-Dec-2003.) Reduce dependencies on axioms. (Revised by BJ, 29-Mar-2020.) |
Ref | Expression |
---|---|
r19.41v | ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | anass 470 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓))) | |
2 | 1 | exbii 1851 | . . 3 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓))) |
3 | 19.41v 1954 | . . 3 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓)) | |
4 | 2, 3 | bitr3i 277 | . 2 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓)) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓)) |
5 | df-rex 3072 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓))) | |
6 | df-rex 3072 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
7 | 6 | anbi1i 625 | . 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ 𝜓) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓)) |
8 | 4, 5, 7 | 3bitr4i 303 | 1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 ∃wex 1782 ∈ wcel 2107 ∃wrex 3071 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-rex 3072 |
This theorem is referenced by: r19.42v 3191 r19.41vv 3225 3reeanv 3228 reuxfr1d 3747 reuind 3750 iuncom4 5006 dfiun2gOLD 5035 iunxiun 5101 inuni 5344 xpiundi 5747 xpiundir 5748 imaco 6251 coiun 6256 abrexco 7243 imaiun 7244 isomin 7334 isoini 7335 imaeqsexv 7360 imaeqexov 7645 oarec 8562 mapsnend 9036 unfi 9172 brttrcl2 9709 genpass 11004 4fvwrd4 13621 4sqlem12 16889 imasleval 17487 lsmspsn 20695 utoptop 23739 metrest 24033 metust 24067 cfilucfil 24068 metuel2 24074 sleadd1 27472 addsuniflem 27484 addsasslem1 27486 addsasslem2 27487 addsdilem1 27606 istrkg2ld 27711 axsegcon 28185 fusgreg2wsp 29589 nmoo0 30044 nmop0 31239 nmfn0 31240 rexunirn 31732 dmrab 31737 iunrnmptss 31797 ressupprn 31912 ordtconnlem1 32904 dya2icoseg2 33277 dya2iocnei 33281 omssubaddlem 33298 omssubadd 33299 satfvsuclem2 34351 satf0 34363 satffunlem1lem2 34394 satffunlem2lem2 34397 bj-mpomptALT 36000 mptsnunlem 36219 fvineqsneq 36293 rabiun 36461 iundif1 36462 poimir 36521 ismblfin 36529 eldmqs1cossres 37529 erimeq2 37548 prter2 37751 prter3 37752 islshpat 37887 lshpsmreu 37979 islpln5 38406 islvol5 38450 cdlemftr3 39436 dvhb1dimN 39857 dib1dim 40036 mapdpglem3 40546 hdmapglem7a 40798 diophrex 41513 r19.41dv 47487 |
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