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Mirrors > Home > MPE Home > Th. List > spcegv | Structured version Visualization version GIF version |
Description: Existential specialization, using implicit substitution. (Contributed by NM, 14-Aug-1994.) Avoid ax-10 2138, ax-11 2155. (Revised by Wolf Lammen, 25-Aug-2023.) |
Ref | Expression |
---|---|
spcgv.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
spcegv | ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∃𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elisset 2816 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) | |
2 | spcgv.1 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 2 | biimprcd 249 | . . 3 ⊢ (𝜓 → (𝑥 = 𝐴 → 𝜑)) |
4 | 3 | eximdv 1921 | . 2 ⊢ (𝜓 → (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝜑)) |
5 | 1, 4 | syl5com 31 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∃𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∃wex 1782 ∈ wcel 2107 |
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-8 2109 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-clel 2811 |
This theorem is referenced by: spcedv 3589 spcev 3597 eqeu 3703 absneu 4733 issn 4834 elpreqprlem 4867 elunii 4914 axpweq 5349 brcogw 5869 opeldmd 5907 breldmg 5910 dmsnopg 6213 predtrss 6324 fvrnressn 7159 f1oexbi 7919 unielxp 8013 frrlem13 8283 f1oen4g 8960 f1dom4g 8961 f1oen3g 8962 f1dom3g 8963 f1domg 8968 en2sn 9041 en2snOLD 9042 en2prd 9048 fodomr 9128 ordiso 9511 fowdom 9566 inf0 9616 infeq5i 9631 oncard 9955 cardsn 9964 dfac8b 10026 ac10ct 10029 aceq3lem 10115 dfacacn 10136 cflem 10241 cflecard 10248 cfslb 10261 coftr 10268 alephsing 10271 fin4i 10293 axdc4lem 10450 gchi 10619 hasheqf1oi 14311 relexpindlem 15010 climeu 15499 brcici 17747 initoeu2lem2 17965 gsumval2a 18604 uptx 23129 alexsubALTlem1 23551 ptcmplem3 23558 prdsxmslem2 24038 tgjustf 27724 tgjustr 27725 wlksnwwlknvbij 29162 clwwlkvbij 29366 aciunf1lem 31887 locfinref 32821 fnimage 34901 fnessref 35242 refssfne 35243 filnetlem4 35266 bj-restb 35975 fin2so 36475 unirep 36582 indexa 36601 nssd 43794 choicefi 43899 axccdom 43921 stoweidlem5 44721 stoweidlem27 44743 stoweidlem28 44744 stoweidlem31 44747 stoweidlem43 44759 stoweidlem44 44760 stoweidlem51 44767 stoweidlem59 44775 nsssmfmbflem 45494 fundcmpsurinjpreimafv 46076 sprbisymrel 46167 isomuspgrlem2 46501 uspgrbisymrelALT 46533 irinitoringc 46967 |
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