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Theorem prv 34488
Description: The "proves" relation on a set. A wff encoded as 𝑈 is true in a model 𝑀 iff for every valuation 𝑠 ∈ (𝑀m ω), the interpretation of the wff using the membership relation on 𝑀 is true. (Contributed by AV, 5-Nov-2023.)
Assertion
Ref Expression
prv ((𝑀𝑉𝑈𝑊) → (𝑀𝑈 ↔ (𝑀 Sat 𝑈) = (𝑀m ω)))

Proof of Theorem prv
Dummy variables 𝑚 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq12 7420 . . 3 ((𝑚 = 𝑀𝑢 = 𝑈) → (𝑚 Sat 𝑢) = (𝑀 Sat 𝑈))
2 simpl 483 . . . 4 ((𝑚 = 𝑀𝑢 = 𝑈) → 𝑚 = 𝑀)
32oveq1d 7426 . . 3 ((𝑚 = 𝑀𝑢 = 𝑈) → (𝑚m ω) = (𝑀m ω))
41, 3eqeq12d 2748 . 2 ((𝑚 = 𝑀𝑢 = 𝑈) → ((𝑚 Sat 𝑢) = (𝑚m ω) ↔ (𝑀 Sat 𝑈) = (𝑀m ω)))
5 df-prv 34406 . 2 ⊧ = {⟨𝑚, 𝑢⟩ ∣ (𝑚 Sat 𝑢) = (𝑚m ω)}
64, 5brabga 5534 1 ((𝑀𝑉𝑈𝑊) → (𝑀𝑈 ↔ (𝑀 Sat 𝑈) = (𝑀m ω)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106   class class class wbr 5148  (class class class)co 7411  ωcom 7857  m cmap 8822   Sat csate 34398  cprv 34399
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-iota 6495  df-fv 6551  df-ov 7414  df-prv 34406
This theorem is referenced by:  elnanelprv  34489  prv0  34490  prv1n  34491
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