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Theorem spcimgf 2810
Description: Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
spcimgf.1 𝑥𝐴
spcimgf.2 𝑥𝜓
spcimgf.3 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
spcimgf (𝐴𝑉 → (∀𝑥𝜑𝜓))

Proof of Theorem spcimgf
StepHypRef Expression
1 spcimgf.2 . . 3 𝑥𝜓
2 spcimgf.1 . . 3 𝑥𝐴
31, 2spcimgft 2806 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝑉 → (∀𝑥𝜑𝜓)))
4 spcimgf.3 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
53, 4mpg 1444 1 (𝐴𝑉 → (∀𝑥𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1346   = wceq 1348  wnf 1453  wcel 2141  wnfc 2299
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732
This theorem is referenced by:  bj-nn0sucALT  13973
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