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Theorem spcimgf 3550
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, 2spcimgfi1 3547 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝑉 → (∀𝑥𝜑𝜓)))
4 spcimgf.3 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
53, 4mpg 1797 1 (𝐴𝑉 → (∀𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1538   = wceq 1540  wnf 1783  wcel 2108  wnfc 2890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1780  df-nf 1784  df-cleq 2729  df-clel 2816  df-nfc 2892
This theorem is referenced by:  spcimegf  3551  iooelexlt  37363
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