Theorem goaleq12d 35416

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 35407	. . 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 4832	. . . 4 ⊢ (𝜑 → ⟨𝑀, 𝐴⟩ = ⟨𝑁, 𝐵⟩)
6	5	opeq2d 4831	. . 3 ⊢ (𝜑 → ⟨2o, ⟨𝑀, 𝐴⟩⟩ = ⟨2o, ⟨𝑁, 𝐵⟩⟩)
7		df-goal 35407	. . . . 5 ⊢ ∀𝑔𝑁𝐵 = ⟨2o, ⟨𝑁, 𝐵⟩⟩
8	7	eqcomi 2742	. . . 4 ⊢ ⟨2o, ⟨𝑁, 𝐵⟩⟩ = ∀𝑔𝑁𝐵
9	8	a1i 11	. . 3 ⊢ (𝜑 → ⟨2o, ⟨𝑁, 𝐵⟩⟩ = ∀𝑔𝑁𝐵)
10	6, 9	eqtrd 2768	. 2 ⊢ (𝜑 → ⟨2o, ⟨𝑀, 𝐴⟩⟩ = ∀𝑔𝑁𝐵)
11	2, 10	eqtrd 2768	1 ⊢ (𝜑 → ∀𝑔𝑀𝐴 = ∀𝑔𝑁𝐵)

Syntax hints:   → wi 4   = wceq 1541  ⟨cop 4581  2_oc2o 8385  ∀_𝑔cgol 35400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-goal 35407

This theorem is referenced by:  satfv1  35428  satfdmlem  35433  fmlasuc  35451  fmla1  35452  satffunlem1lem1  35467  satffunlem1lem2  35468  satffunlem2lem1  35469  satffunlem2lem2  35471  satfv1fvfmla1  35488  2goelgoanfmla1  35489