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Theorem spcgv 2825
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 2319 . 2 𝑥𝐴
2 nfv 1528 . 2 𝑥𝜓
3 spcgv.1 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
41, 2, 3spcgf 2820 1 (𝐴𝑉 → (∀𝑥𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wal 1351   = wceq 1353  wcel 2148
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 2740
This theorem is referenced by:  spcv  2832  mob2  2918  intss1  3860  dfiin2g  3920  exmidsssnc  4204  exmid1stab  4209  frirrg  4351  frind  4353  alxfr  4462  elirr  4541  en2lp  4554  tfisi  4587  mptfvex  5602  tfrcl  6365  rdgisucinc  6386  frecabex  6399  fisseneq  6931  mkvprop  7156  exmidfodomrlemr  7201  exmidfodomrlemrALT  7202  acfun  7206  exmidmotap  7260  ccfunen  7263  zfz1isolem1  10820  zfz1iso  10821  uniopn  13504  pw1nct  14755  sbthom  14777
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