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Theorem spcgf 2819
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 2814 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝑉 → (∀𝑥𝜑𝜓)))
4 spcgf.3 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
53, 4mpg 1451 1 (𝐴𝑉 → (∀𝑥𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wal 1351   = wceq 1353  wnf 1460  wcel 2148  wnfc 2306
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 2739
This theorem is referenced by:  spcgv  2824  rspc  2835  elabgt  2878  eusvnf  4453  mpofvex  6203
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