![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2139, 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 2821 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) | |
2 | spcgv.1 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 2 | biimprcd 250 | . . 3 ⊢ (𝜓 → (𝑥 = 𝐴 → 𝜑)) |
4 | 3 | eximdv 1915 | . 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 206 = wceq 1537 ∃wex 1776 ∈ wcel 2106 |
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 ax-6 1965 ax-7 2005 ax-8 2108 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-clel 2814 |
This theorem is referenced by: spcedv 3598 spcev 3606 eqeu 3715 absneu 4733 issn 4837 elpreqprlem 4871 elunii 4917 axpweq 5357 brcogw 5882 opeldmd 5920 breldmg 5923 dmsnopg 6235 predtrss 6345 fvrnressn 7181 f1oexbi 7951 unielxp 8051 frrlem13 8322 f1oen4g 9004 f1dom4g 9005 f1oen3g 9006 f1dom3g 9007 f1domg 9011 en2sn 9080 en2prd 9087 fodomr 9167 fodomfir 9366 ordiso 9554 fowdom 9609 inf0 9659 infeq5i 9674 oncard 9998 cardsn 10007 dfac8b 10069 ac10ct 10072 aceq3lem 10158 dfacacn 10180 cflem 10283 cflemOLD 10284 cflecard 10291 cfslb 10304 coftr 10311 alephsing 10314 fin4i 10336 axdc4lem 10493 gchi 10662 hasheqf1oi 14387 relexpindlem 15099 climeu 15588 brcici 17848 initoeu2lem2 18069 gsumval2a 18711 irinitoringc 21508 uptx 23649 alexsubALTlem1 24071 ptcmplem3 24078 prdsxmslem2 24558 tgjustf 28496 tgjustr 28497 wlksnwwlknvbij 29938 clwwlkvbij 30142 aciunf1lem 32679 locfinref 33802 fnimage 35911 fnessref 36340 refssfne 36341 filnetlem4 36364 bj-restb 37077 fin2so 37594 unirep 37701 indexa 37720 nssd 45045 choicefi 45143 axccdom 45165 stoweidlem5 45961 stoweidlem27 45983 stoweidlem28 45984 stoweidlem31 45987 stoweidlem43 45999 stoweidlem44 46000 stoweidlem51 46007 stoweidlem59 46015 nsssmfmbflem 46734 fundcmpsurinjpreimafv 47333 sprbisymrel 47424 uspgrbisymrelALT 47999 |
Copyright terms: Public domain | W3C validator |