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Theorem spcv 2831
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 2824 . 2 (𝐴 ∈ V → (∀𝑥𝜑𝜓))
41, 3ax-mp 5 1 (∀𝑥𝜑𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wal 1351   = wceq 1353  wcel 2148  Vcvv 2737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739
This theorem is referenced by:  morex  2921  exmidexmid  4193  exmidsssn  4199  exmidel  4202  rext  4211  ontr2exmid  4520  regexmidlem1  4528  reg2exmid  4531  relop  4772  disjxp1  6230  rdgtfr  6368  ssfiexmid  6869  domfiexmid  6871  diffitest  6880  findcard  6881  fiintim  6921  fisseneq  6924  finomni  7131  exmidomni  7133  exmidlpo  7134  exmidunben  12397  bj-d0clsepcl  14299  bj-inf2vnlem1  14344  subctctexmid  14373
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