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Theorem prv 35412
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 7439 . . 3 ((𝑚 = 𝑀𝑢 = 𝑈) → (𝑚 Sat 𝑢) = (𝑀 Sat 𝑈))
2 simpl 482 . . . 4 ((𝑚 = 𝑀𝑢 = 𝑈) → 𝑚 = 𝑀)
32oveq1d 7445 . . 3 ((𝑚 = 𝑀𝑢 = 𝑈) → (𝑚m ω) = (𝑀m ω))
41, 3eqeq12d 2750 . 2 ((𝑚 = 𝑀𝑢 = 𝑈) → ((𝑚 Sat 𝑢) = (𝑚m ω) ↔ (𝑀 Sat 𝑈) = (𝑀m ω)))
5 df-prv 35330 . 2 ⊧ = {⟨𝑚, 𝑢⟩ ∣ (𝑚 Sat 𝑢) = (𝑚m ω)}
64, 5brabga 5543 1 ((𝑀𝑉𝑈𝑊) → (𝑀𝑈 ↔ (𝑀 Sat 𝑈) = (𝑀m ω)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105   class class class wbr 5147  (class class class)co 7430  ωcom 7886  m cmap 8864   Sat csate 35322  cprv 35323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-iota 6515  df-fv 6570  df-ov 7433  df-prv 35330
This theorem is referenced by:  elnanelprv  35413  prv0  35414  prv1n  35415
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