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Mirrors > Home > MPE Home > Th. List > exlimi | Structured version Visualization version GIF version |
Description: Inference associated with 19.23 2205. See exlimiv 1934 for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 10-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) |
Ref | Expression |
---|---|
exlimi.1 | ⊢ Ⅎ𝑥𝜓 |
exlimi.2 | ⊢ (𝜑 → 𝜓) |
Ref | Expression |
---|---|
exlimi | ⊢ (∃𝑥𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exlimi.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
2 | 1 | 19.23 2205 | . 2 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)) |
3 | exlimi.2 | . 2 ⊢ (𝜑 → 𝜓) | |
4 | 2, 3 | mpgbi 1801 | 1 ⊢ (∃𝑥𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1782 Ⅎwnf 1786 |
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 ax-6 1972 ax-7 2012 ax-12 2172 |
This theorem depends on definitions: df-bi 206 df-ex 1783 df-nf 1787 |
This theorem is referenced by: sbalex 2236 equsexv 2260 equs5av 2271 exlimih 2286 equs5aALT 2364 equs5eALT 2365 equsex 2418 exdistrf 2447 equs5a 2457 equs5e 2458 dfmoeu 2531 moanim 2617 euan 2618 moexexlem 2623 2eu6 2653 ceqsexOLD 3525 sbhypfOLD 3540 vtoclef 3547 vtoclgf 3555 vtoclg1f 3556 rspn0OLD 4354 reusv2lem1 5397 copsexgw 5491 copsexg 5492 copsex2gOLD 5495 rexopabb 5529 0nelopabOLD 5569 ralxpf 5847 dmcoss 5971 fv3 6910 opabiota 6975 oprabidw 7440 zfregcl 9589 scottex 9880 scott0 9881 dfac5lem5 10122 zfcndpow 10611 zfcndreg 10612 zfcndinf 10613 reclem2pr 11043 mreiincl 17540 brabgaf 31837 cnvoprabOLD 31945 bnj607 33927 bnj900 33940 exisym1 35309 exlimii 35709 bj-exlimmpi 35792 bj-exlimmpbi 35793 bj-exlimmpbir 35794 dihglblem5 40169 tworepnotupword 45600 eu2ndop1stv 45833 pgind 47762 |
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