Theorem rexabso 45565

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 5229	. . 3 ⊢ (Tr 𝑀 → (𝐴 ∈ 𝑀 → 𝐴 ⊆ 𝑀))
2	1	imp 411	. 2 ⊢ ((Tr 𝑀 ∧ 𝐴 ∈ 𝑀) → 𝐴 ⊆ 𝑀)
3		rexss 4019	. 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 2149  ∃wrex 3095   ⊆ wss 3913  Tr wtr 5219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-v 3465  df-ss 3930  df-uni 4874  df-tr 5220

This theorem is referenced by:  rexabsod  45567  n0abso  45572