Theorem ralabsod 45566

Description: Deduction form of ralabso 45564. (Contributed by Eric Schmidt, 19-Oct-2025.)

Hypothesis

Ref	Expression
ralabsod.1	⊢ (𝜑 → Tr 𝑀)

Assertion

Ref	Expression
ralabsod	⊢ ((𝜑 ∧ 𝐴 ∈ 𝑀) → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 → 𝜓)))

Distinct variable groups:   𝑥,𝑀   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem ralabsod

Step	Hyp	Ref	Expression
1		ralabsod.1	. 2 ⊢ (𝜑 → Tr 𝑀)
2		ralabso 45564	. 2 ⊢ ((Tr 𝑀 ∧ 𝐴 ∈ 𝑀) → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 → 𝜓)))
3	1, 2	sylan 591	1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑀) → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 → 𝜓)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2149  ∀wral 3085  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-v 3465  df-ss 3930  df-uni 4874  df-tr 5220

This theorem is referenced by:  ralabsobidv  45568