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Theorem spcgv 2745
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 2256 . 2 𝑥𝐴
2 nfv 1491 . 2 𝑥𝜓
3 spcgv.1 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
41, 2, 3spcgf 2740 1 (𝐴𝑉 → (∀𝑥𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1312   = wceq 1314  wcel 1463
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660
This theorem is referenced by:  spcv  2751  mob2  2835  intss1  3754  dfiin2g  3814  exmidsssnc  4094  frirrg  4240  frind  4242  alxfr  4350  elirr  4424  en2lp  4437  tfisi  4469  mptfvex  5472  tfrcl  6227  rdgisucinc  6248  frecabex  6261  fisseneq  6786  mkvprop  6998  exmidfodomrlemr  7022  exmidfodomrlemrALT  7023  acfun  7027  ccfunen  7043  zfz1isolem1  10523  zfz1iso  10524  uniopn  12063  exmid1stab  12997  sbthom  13023
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