Theorem goaleq12d 33292

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 33283	. . 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 4817	. . . 4 ⊢ (𝜑 → ⟨𝑀, 𝐴⟩ = ⟨𝑁, 𝐵⟩)
6	5	opeq2d 4816	. . 3 ⊢ (𝜑 → ⟨2o, ⟨𝑀, 𝐴⟩⟩ = ⟨2o, ⟨𝑁, 𝐵⟩⟩)
7		df-goal 33283	. . . . 5 ⊢ ∀𝑔𝑁𝐵 = ⟨2o, ⟨𝑁, 𝐵⟩⟩
8	7	eqcomi 2748	. . . 4 ⊢ ⟨2o, ⟨𝑁, 𝐵⟩⟩ = ∀𝑔𝑁𝐵
9	8	a1i 11	. . 3 ⊢ (𝜑 → ⟨2o, ⟨𝑁, 𝐵⟩⟩ = ∀𝑔𝑁𝐵)
10	6, 9	eqtrd 2779	. 2 ⊢ (𝜑 → ⟨2o, ⟨𝑀, 𝐴⟩⟩ = ∀𝑔𝑁𝐵)
11	2, 10	eqtrd 2779	1 ⊢ (𝜑 → ∀𝑔𝑀𝐴 = ∀𝑔𝑁𝐵)

Syntax hints:   → wi 4   = wceq 1541  ⟨cop 4572  2_oc2o 8275  ∀_𝑔cgol 33276

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-goal 33283

This theorem is referenced by:  satfv1  33304  satfdmlem  33309  fmlasuc  33327  fmla1  33328  satffunlem1lem1  33343  satffunlem1lem2  33344  satffunlem2lem1  33345  satffunlem2lem2  33347  satfv1fvfmla1  33364  2goelgoanfmla1  33365