Theorem ralabso 45710

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 5230	. . 3 ⊢ (Tr 𝑀 → (𝐴 ∈ 𝑀 → 𝐴 ⊆ 𝑀))
2	1	imp 412	. 2 ⊢ ((Tr 𝑀 ∧ 𝐴 ∈ 𝑀) → 𝐴 ⊆ 𝑀)
3		ralss 4011	. 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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2146  ∀wral 3081   ⊆ wss 3906  Tr wtr 5220

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-v 3459  df-ss 3923  df-uni 4875  df-tr 5221

This theorem is used by:  ralabsod  45712  ssabso  45716  disjabso  45717  modelac8prim  45734