Theorem goaleq12d 36296

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 36287	. . 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 4851	. . . 4 ⊢ (𝜑 → ⟨𝑀, 𝐴⟩ = ⟨𝑁, 𝐵⟩)
6	5	opeq2d 4850	. . 3 ⊢ (𝜑 → ⟨2o, ⟨𝑀, 𝐴⟩⟩ = ⟨2o, ⟨𝑁, 𝐵⟩⟩)
7		df-goal 36287	. . . . 5 ⊢ ∀𝑔𝑁𝐵 = ⟨2o, ⟨𝑁, 𝐵⟩⟩
8	7	eqcomi 2772	. . . 4 ⊢ ⟨2o, ⟨𝑁, 𝐵⟩⟩ = ∀𝑔𝑁𝐵
9	8	a1i 11	. . 3 ⊢ (𝜑 → ⟨2o, ⟨𝑁, 𝐵⟩⟩ = ∀𝑔𝑁𝐵)
10	6, 9	eqtrd 2798	. 2 ⊢ (𝜑 → ⟨2o, ⟨𝑀, 𝐴⟩⟩ = ∀𝑔𝑁𝐵)
11	2, 10	eqtrd 2798	1 ⊢ (𝜑 → ∀𝑔𝑀𝐴 = ∀𝑔𝑁𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  ⟨cop 4838  2_oc2o 8501  ∀_𝑔cgol 36280

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-goal 36287

This theorem is used by:  satfv1  36308  satfdmlem  36313  fmlasuc  36331  fmla1  36332  satffunlem1lem1  36347  satffunlem1lem2  36348  satffunlem2lem1  36349  satffunlem2lem2  36351  satfv1fvfmla1  36368  2goelgoanfmla1  36369