Theorem goaleq12d 35714

Description: Equality of the "Godel-set of universal quantification". (Contributed by AV, 18-Sep-2023.)

Hypotheses

Ref	Expression
goaleq12d.1	⊢ (𝜑 → 𝑀 = 𝑁)
goaleq12d.2	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
goaleq12d	⊢ (𝜑 → ∀𝑔𝑀𝐴 = ∀𝑔𝑁𝐵)

Proof of Theorem goaleq12d

Step	Hyp	Ref	Expression
1		df-goal 35705	. . 3 ⊢ ∀𝑔𝑀𝐴 = ⟨2o, ⟨𝑀, 𝐴⟩⟩
2	1	a1i 11	. 2 ⊢ (𝜑 → ∀𝑔𝑀𝐴 = ⟨2o, ⟨𝑀, 𝐴⟩⟩)
3		goaleq12d.1	. . . . 5 ⊢ (𝜑 → 𝑀 = 𝑁)
4		goaleq12d.2	. . . . 5 ⊢ (𝜑 → 𝐴 = 𝐵)
5	3, 4	opeq12d 4842	. . . 4 ⊢ (𝜑 → ⟨𝑀, 𝐴⟩ = ⟨𝑁, 𝐵⟩)
6	5	opeq2d 4841	. . 3 ⊢ (𝜑 → ⟨2o, ⟨𝑀, 𝐴⟩⟩ = ⟨2o, ⟨𝑁, 𝐵⟩⟩)
7		df-goal 35705	. . . . 5 ⊢ ∀𝑔𝑁𝐵 = ⟨2o, ⟨𝑁, 𝐵⟩⟩
8	7	eqcomi 2774	. . . 4 ⊢ ⟨2o, ⟨𝑁, 𝐵⟩⟩ = ∀𝑔𝑁𝐵
9	8	a1i 11	. . 3 ⊢ (𝜑 → ⟨2o, ⟨𝑁, 𝐵⟩⟩ = ∀𝑔𝑁𝐵)
10	6, 9	eqtrd 2800	. 2 ⊢ (𝜑 → ⟨2o, ⟨𝑀, 𝐴⟩⟩ = ∀𝑔𝑁𝐵)
11	2, 10	eqtrd 2800	1 ⊢ (𝜑 → ∀𝑔𝑀𝐴 = ∀𝑔𝑁𝐵)

Syntax hints:   → wi 4   = wceq 1563  ⟨cop 4591  2_oc2o 8435  ∀_𝑔cgol 35698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-goal 35705

This theorem is referenced by:  satfv1  35726  satfdmlem  35731  fmlasuc  35749  fmla1  35750  satffunlem1lem1  35765  satffunlem1lem2  35766  satffunlem2lem1  35767  satffunlem2lem2  35769  satfv1fvfmla1  35786  2goelgoanfmla1  35787