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Theorem rspcdva 2708
Description: Restricted specialization, using implicit substitution. (Contributed by Thierry Arnoux, 21-Jun-2020.)
Hypotheses
Ref Expression
rspcdva.1 (𝑥 = 𝐶 → (𝜓𝜒))
rspcdva.2 (𝜑 → ∀𝑥𝐴 𝜓)
rspcdva.3 (𝜑𝐶𝐴)
Assertion
Ref Expression
rspcdva (𝜑𝜒)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝜒,𝑥   𝜑,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem rspcdva
StepHypRef Expression
1 rspcdva.2 . 2 (𝜑 → ∀𝑥𝐴 𝜓)
2 rspcdva.3 . . 3 (𝜑𝐶𝐴)
3 rspcdva.1 . . . 4 (𝑥 = 𝐶 → (𝜓𝜒))
43adantl 271 . . 3 ((𝜑𝑥 = 𝐶) → (𝜓𝜒))
52, 4rspcdv 2705 . 2 (𝜑 → (∀𝑥𝐴 𝜓𝜒))
61, 5mpd 13 1 (𝜑𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1285  wcel 1434  wral 2349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-v 2604
This theorem is referenced by:  tfr1onlemsucfn  5989  tfr1onlemsucaccv  5990  tfr1onlembxssdm  5992  tfr1onlembfn  5993  tfr1onlemres  5998  tfrcllemsucfn  6002  tfrcllemsucaccv  6003  tfrcllembxssdm  6005  tfrcllembfn  6006  tfrcllemres  6011  tfrcl  6013  rdgon  6035  frecabcl  6048  exbtwnzlemstep  9334  frecuzrdgsuc  9496  frecuzrdgg  9498  frecuzrdgsuctlem  9505  uzsinds  9518  iseqvalt  9532  iseq1  9533  iseq1t  9534  iseqfcl  9535  iseqfclt  9536  iseqcl  9537  iseqp1  9538  iseqp1t  9539  iseqoveq  9540  iseqfveq2  9544  iseqfveq  9546  iseqshft2  9548  isermono  9553  iseqsplit  9554  iseqcaopr3  9556  iseqid3s  9562  iseqid  9563  iseqid2  9564  iseqhomo  9565  iseqz  9566  isumrblem  10337  fisumcvg  10338  bezoutlemex  10534  bezoutlemzz  10535  bezoutlemmo  10539  bezoutlemle  10541  bezoutlemsup  10542
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