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Theorem prv 32702
 Description: The "proves" relation on a set. A wff encoded as 𝑈 is true in a model 𝑀 iff for every valuation 𝑠 ∈ (𝑀 ↑m 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 7155 . . 3 ((𝑚 = 𝑀𝑢 = 𝑈) → (𝑚 Sat 𝑢) = (𝑀 Sat 𝑈))
2 simpl 486 . . . 4 ((𝑚 = 𝑀𝑢 = 𝑈) → 𝑚 = 𝑀)
32oveq1d 7161 . . 3 ((𝑚 = 𝑀𝑢 = 𝑈) → (𝑚m ω) = (𝑀m ω))
41, 3eqeq12d 2840 . 2 ((𝑚 = 𝑀𝑢 = 𝑈) → ((𝑚 Sat 𝑢) = (𝑚m ω) ↔ (𝑀 Sat 𝑈) = (𝑀m ω)))
5 df-prv 32620 . 2 ⊧ = {⟨𝑚, 𝑢⟩ ∣ (𝑚 Sat 𝑢) = (𝑚m ω)}
64, 5brabga 5409 1 ((𝑀𝑉𝑈𝑊) → (𝑀𝑈 ↔ (𝑀 Sat 𝑈) = (𝑀m ω)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115   class class class wbr 5053  (class class class)co 7146  ωcom 7571   ↑m cmap 8398   Sat∈ csate 32612  ⊧cprv 32613 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-opab 5116  df-iota 6303  df-fv 6352  df-ov 7149  df-prv 32620 This theorem is referenced by:  elnanelprv  32703  prv0  32704  prv1n  32705
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