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Theorem prv 35433
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 7440 . . 3 ((𝑚 = 𝑀𝑢 = 𝑈) → (𝑚 Sat 𝑢) = (𝑀 Sat 𝑈))
2 simpl 482 . . . 4 ((𝑚 = 𝑀𝑢 = 𝑈) → 𝑚 = 𝑀)
32oveq1d 7446 . . 3 ((𝑚 = 𝑀𝑢 = 𝑈) → (𝑚m ω) = (𝑀m ω))
41, 3eqeq12d 2753 . 2 ((𝑚 = 𝑀𝑢 = 𝑈) → ((𝑚 Sat 𝑢) = (𝑚m ω) ↔ (𝑀 Sat 𝑈) = (𝑀m ω)))
5 df-prv 35351 . 2 ⊧ = {⟨𝑚, 𝑢⟩ ∣ (𝑚 Sat 𝑢) = (𝑚m ω)}
64, 5brabga 5539 1 ((𝑀𝑉𝑈𝑊) → (𝑀𝑈 ↔ (𝑀 Sat 𝑈) = (𝑀m ω)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108   class class class wbr 5143  (class class class)co 7431  ωcom 7887  m cmap 8866   Sat csate 35343  cprv 35344
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-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-iota 6514  df-fv 6569  df-ov 7434  df-prv 35351
This theorem is referenced by:  elnanelprv  35434  prv0  35435  prv1n  35436
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