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Theorem spcgf 2702
Description: Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 2-Feb-1997.) (Revised by Andrew Salmon, 12-Aug-2011.)
Hypotheses
Ref Expression
spcgf.1 𝑥𝐴
spcgf.2 𝑥𝜓
spcgf.3 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
spcgf (𝐴𝑉 → (∀𝑥𝜑𝜓))

Proof of Theorem spcgf
StepHypRef Expression
1 spcgf.2 . . 3 𝑥𝜓
2 spcgf.1 . . 3 𝑥𝐴
31, 2spcgft 2697 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝑉 → (∀𝑥𝜑𝜓)))
4 spcgf.3 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
53, 4mpg 1386 1 (𝐴𝑉 → (∀𝑥𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1288   = wceq 1290  wnf 1395  wcel 1439  wnfc 2216
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622
This theorem is referenced by:  spcgv  2707  rspc  2717  elabgt  2758  eusvnf  4288  mpt2fvex  5987
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