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Mirrors > Home > MPE Home > Th. List > spcedv | Structured version Visualization version GIF version |
Description: Existential specialization, using implicit substitution, deduction version. (Contributed by RP, 12-Aug-2020.) (Revised by AV, 16-Aug-2024.) |
Ref | Expression |
---|---|
spcedv.1 | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
spcedv.2 | ⊢ (𝜑 → 𝜒) |
spcedv.3 | ⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
spcedv | ⊢ (𝜑 → ∃𝑥𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spcedv.1 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
2 | spcedv.2 | . 2 ⊢ (𝜑 → 𝜒) | |
3 | spcedv.3 | . . 3 ⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜒)) | |
4 | 3 | spcegv 3588 | . 2 ⊢ (𝑋 ∈ 𝑉 → (𝜒 → ∃𝑥𝜓)) |
5 | 1, 2, 4 | sylc 65 | 1 ⊢ (𝜑 → ∃𝑥𝜓) |
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: selsALT 5440 zfrep6 7941 wfrlem15OLD 8323 ertr 8718 f1dom2gOLD 8966 dom3d 8990 disjenex 9135 domssex2 9137 domssex 9138 brwdom2 9568 infxpenc2lem2 10015 dfac8clem 10027 ac5num 10031 acni2 10041 acnlem 10043 finnisoeu 10108 infpss 10212 cofsmo 10264 axdc4lem 10450 ac6num 10474 axdclem2 10515 hasheqf1od 14313 fz1isolem 14422 wrd2f1tovbij 14911 fsum 15666 ntrivcvgn0 15844 fprod 15885 setsexstruct2 17108 isacs1i 17601 mreacs 17602 gsumval3lem2 19774 eltg3 22465 elptr 23077 nbusgrf1o1 28658 cusgrexg 28732 cusgrfilem3 28745 sizusglecusg 28751 wwlksnextbij 29187 gsumhashmul 32239 bj-imdirco 36119 eqvreltr 37525 sticksstones20 41030 onsucf1lem 42067 onsucf1olem 42068 nnoeomeqom 42110 rp-isfinite5 42316 clrellem 42421 clcnvlem 42422 fundcmpsurinj 46125 prproropen 46224 isomgreqve 46541 isomushgr 46542 isomgrsym 46552 isomgrtr 46555 ushrisomgr 46557 uspgrsprfo 46574 uspgrbispr 46577 1aryenef 47379 2aryenef 47390 eufsnlem 47555 opncldeqv 47582 thincciso 47717 mndtcbas 47755 |
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