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Mirrors > Home > MPE Home > Th. List > 2exbii | Structured version Visualization version GIF version |
Description: Inference adding two existential quantifiers to both sides of an equivalence. (Contributed by NM, 16-Mar-1995.) |
Ref | Expression |
---|---|
2exbii.1 | ⊢ (𝜑 ↔ 𝜓) |
Ref | Expression |
---|---|
2exbii | ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2exbii.1 | . . 3 ⊢ (𝜑 ↔ 𝜓) | |
2 | 1 | exbii 1851 | . 2 ⊢ (∃𝑦𝜑 ↔ ∃𝑦𝜓) |
3 | 2 | exbii 1851 | 1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∃wex 1782 |
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-ex 1783 |
This theorem is referenced by: 3exbii 1853 2exanali 1864 4exdistrv 1961 3exdistr 1965 cbvex4vw 2046 eeeanv 2347 ee4anv 2348 2exsb 2357 cbvex4v 2415 2sb5rf 2472 sbel2x 2474 2mo2 2644 reeanlem 3226 rexcomf 3301 cgsex4g 3521 ceqsex3v 3532 ceqsex4v 3533 ceqsex8v 3535 copsexgw 5491 copsexg 5492 copsex2g 5494 vopelopabsb 5530 opabn0 5554 elxp2 5701 rabxp 5725 elxp3 5743 elvv 5751 elvvv 5752 copsex2gb 5807 elcnv2 5878 cnvuni 5887 xpdifid 6168 coass 6265 fununi 6624 dfmpt3 6685 tpres 7202 dfoprab2 7467 dmoprab 7510 rnoprab 7512 mpomptx 7521 resoprab 7526 elrnmpores 7546 ov3 7570 ov6g 7571 uniuni 7749 opabex3rd 7953 oprabex3 7964 wfrlem4OLD 8312 oeeu 8603 xpassen 9066 sbthfilem 9201 zorn2lem6 10496 ltresr 11135 axaddf 11140 axmulf 11141 hashfun 14397 hash2prb 14433 5oalem7 30944 mpomptxf 31936 eulerpartlemgvv 33406 bnj600 33961 bnj916 33975 bnj983 33993 bnj986 33997 bnj996 33998 bnj1021 34008 dfacycgr1 34166 satfv1 34385 elima4 34778 brtxp2 34884 brpprod3a 34889 brpprod3b 34890 elfuns 34918 brcart 34935 brimg 34940 brapply 34941 brsuccf 34944 brrestrict 34952 dfrdg4 34954 ellines 35155 bj-cbvex4vv 35731 itg2addnclem3 36589 brxrn2 37293 dfxrn2 37294 ecxrn 37305 inxpxrn 37313 rnxrn 37316 dalem20 38612 dvhopellsm 40036 diblsmopel 40090 ralopabb 42210 en2pr 42346 pm11.52 43194 pm11.6 43199 pm11.7 43203 opelopab4 43360 stoweidlem35 44799 fundcmpsurbijinj 46126 mpomptx2 47058 |
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