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Theorem spcimgf 3506
 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 3504 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝑉 → (∀𝑥𝜑𝜓)))
4 spcimgf.3 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
53, 4mpg 1799 1 (𝐴𝑉 → (∀𝑥𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1536   = wceq 1538  Ⅎwnf 1785   ∈ wcel 2111  Ⅎwnfc 2899 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-v 3411 This theorem is referenced by:  spcimegf  3507  iooelexlt  35081
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