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Theorem spcv 2784
 Description: Rule of specialization, using implicit substitution. (Contributed by NM, 22-Jun-1994.)
Hypotheses
Ref Expression
spcv.1 𝐴 ∈ V
spcv.2 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
spcv (∀𝑥𝜑𝜓)
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem spcv
StepHypRef Expression
1 spcv.1 . 2 𝐴 ∈ V
2 spcv.2 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
32spcgv 2777 . 2 (𝐴 ∈ V → (∀𝑥𝜑𝜓))
41, 3ax-mp 5 1 (∀𝑥𝜑𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1330   = wceq 1332   ∈ wcel 1481  Vcvv 2690 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2692 This theorem is referenced by:  morex  2873  exmidexmid  4129  exmidsssn  4134  exmidel  4137  rext  4146  ontr2exmid  4449  regexmidlem1  4457  reg2exmid  4460  relop  4699  disjxp1  6143  rdgtfr  6281  ssfiexmid  6780  domfiexmid  6782  diffitest  6791  findcard  6792  fiintim  6830  fisseneq  6833  finomni  7025  exmidomni  7027  exmidlpo  7028  exmidunben  11998  bj-d0clsepcl  13327  bj-inf2vnlem1  13372  subctctexmid  13401
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