Theorem rexabso 45706

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
rexabso	⊢ ((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 rexabso

Step	Hyp	Ref	Expression
1		trss 5227	. . 3 ⊢ (Tr 𝑀 → (𝐴 ∈ 𝑀 → 𝐴 ⊆ 𝑀))
2	1	imp 411	. 2 ⊢ ((Tr 𝑀 ∧ 𝐴 ∈ 𝑀) → 𝐴 ⊆ 𝑀)
3		rexss 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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2142  ∃wrex 3088   ⊆ wss 3904  Tr wtr 5217

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-v 3456  df-ss 3921  df-uni 4872  df-tr 5218

This theorem is used by:  rexabsod  45708  n0abso  45713