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Theorem spcgv 3279
Description: Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 22-Jun-1994.)
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 nfcv 2761 . 2 𝑥𝐴
2 nfv 1840 . 2 𝑥𝜓
3 spcgv.1 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
41, 2, 3spcgf 3274 1 (𝐴𝑉 → (∀𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1478   = wceq 1480  wcel 1987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188
This theorem is referenced by:  spcv  3285  mob2  3368  intss1  4457  dfiin2g  4519  alxfr  4838  fri  5036  isofrlem  6544  tfisi  7005  limomss  7017  nnlim  7025  f1oweALT  7097  pssnn  8122  findcard3  8147  ttukeylem1  9275  rami  15643  ramcl  15657  islbs3  19074  mplsubglem  19353  mpllsslem  19354  uniopn  20627  chlimi  27937  dfon2lem3  31388  dfon2lem8  31393  neificl  33178  ismrcd1  36738
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