Theorem ralabso 45677

Description: Simplification of restricted quantification in a transitive class. When 𝜑 is quantifier-free, this shows that the formula ∀𝑥 ∈ 𝑦𝜑 is absolute for transitive models, which is a particular case of Lemma I.16.2 of [Kunen2] p. 95. (Contributed by Eric Schmidt, 19-Oct-2025.)

Assertion

Ref	Expression
ralabso	⊢ ((Tr 𝑀 ∧ 𝐴 ∈ 𝑀) → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 → 𝜑)))

Distinct variable groups:   𝑥,𝑀   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ralabso

Step	Hyp	Ref	Expression
1		trss 5228	. . 3 ⊢ (Tr 𝑀 → (𝐴 ∈ 𝑀 → 𝐴 ⊆ 𝑀))
2	1	imp 411	. 2 ⊢ ((Tr 𝑀 ∧ 𝐴 ∈ 𝑀) → 𝐴 ⊆ 𝑀)
3		ralss 4010	. 2 ⊢ (𝐴 ⊆ 𝑀 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 → 𝜑)))
4	2, 3	syl 18	1 ⊢ ((Tr 𝑀 ∧ 𝐴 ∈ 𝑀) → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 → 𝜑)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2143  ∀wral 3079   ⊆ wss 3905  Tr wtr 5218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-v 3457  df-ss 3922  df-uni 4873  df-tr 5219

This theorem is referenced by:  ralabsod  45679  ssabso  45683  disjabso  45684  modelac8prim  45701