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Theorem goaleq12d 35336
Description: Equality of the "Godel-set of universal quantification". (Contributed by AV, 18-Sep-2023.)
Hypotheses
Ref Expression
goaleq12d.1 (𝜑𝑀 = 𝑁)
goaleq12d.2 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
goaleq12d (𝜑 → ∀𝑔𝑀𝐴 = ∀𝑔𝑁𝐵)

Proof of Theorem goaleq12d
StepHypRef Expression
1 df-goal 35327 . . 3 𝑔𝑀𝐴 = ⟨2o, ⟨𝑀, 𝐴⟩⟩
21a1i 11 . 2 (𝜑 → ∀𝑔𝑀𝐴 = ⟨2o, ⟨𝑀, 𝐴⟩⟩)
3 goaleq12d.1 . . . . 5 (𝜑𝑀 = 𝑁)
4 goaleq12d.2 . . . . 5 (𝜑𝐴 = 𝐵)
53, 4opeq12d 4886 . . . 4 (𝜑 → ⟨𝑀, 𝐴⟩ = ⟨𝑁, 𝐵⟩)
65opeq2d 4885 . . 3 (𝜑 → ⟨2o, ⟨𝑀, 𝐴⟩⟩ = ⟨2o, ⟨𝑁, 𝐵⟩⟩)
7 df-goal 35327 . . . . 5 𝑔𝑁𝐵 = ⟨2o, ⟨𝑁, 𝐵⟩⟩
87eqcomi 2744 . . . 4 ⟨2o, ⟨𝑁, 𝐵⟩⟩ = ∀𝑔𝑁𝐵
98a1i 11 . . 3 (𝜑 → ⟨2o, ⟨𝑁, 𝐵⟩⟩ = ∀𝑔𝑁𝐵)
106, 9eqtrd 2775 . 2 (𝜑 → ⟨2o, ⟨𝑀, 𝐴⟩⟩ = ∀𝑔𝑁𝐵)
112, 10eqtrd 2775 1 (𝜑 → ∀𝑔𝑀𝐴 = ∀𝑔𝑁𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  cop 4637  2oc2o 8499  𝑔cgol 35320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-goal 35327
This theorem is referenced by:  satfv1  35348  satfdmlem  35353  fmlasuc  35371  fmla1  35372  satffunlem1lem1  35387  satffunlem1lem2  35388  satffunlem2lem1  35389  satffunlem2lem2  35391  satfv1fvfmla1  35408  2goelgoanfmla1  35409
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