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Theorem spcgv 3586
Description: Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 22-Jun-1994.) Avoid ax-10 2137, ax-11 2154. (Revised by Wolf Lammen, 25-Aug-2023.)
Hypothesis
Ref Expression
spcgv.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
spcgv (𝐴𝑉 → (∀𝑥𝜑𝜓))
Distinct variable groups:   𝜓,𝑥   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem spcgv
StepHypRef Expression
1 elex 3492 . 2 (𝐴𝑉𝐴 ∈ V)
2 id 22 . . 3 (𝐴 ∈ V → 𝐴 ∈ V)
3 spcgv.1 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
43adantl 482 . . 3 ((𝐴 ∈ V ∧ 𝑥 = 𝐴) → (𝜑𝜓))
52, 4spcdv 3584 . 2 (𝐴 ∈ V → (∀𝑥𝜑𝜓))
61, 5syl 17 1 (𝐴𝑉 → (∀𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1539   = wceq 1541  wcel 2106  Vcvv 3474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476
This theorem is referenced by:  spcv  3595  mob2  3711  sbceqal  3843  intss1  4967  dfiin2g  5035  alxfr  5405  friOLD  5637  funmo  6563  isofrlem  7336  tfisi  7847  limomss  7859  nnlim  7868  f1oweALT  7958  pssnn  9167  pssnnOLD  9264  findcard3  9284  findcard3OLD  9285  frmin  9743  ttukeylem1  10503  rami  16947  ramcl  16961  islbs3  20767  mplsubglem  21557  mpllsslem  21558  uniopn  22398  chlimi  30482  iinabrex  31795  dfon2lem3  34752  dfon2lem8  34757  neificl  36616  ismrcd1  41426  mnuop23d  43015
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