Theorem ralabsod 44953

Description: Deduction form of ralabso 44951. (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 44951	. 2 ⊢ ((Tr 𝑀 ∧ 𝐴 ∈ 𝑀) → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 → 𝜓)))
3	1, 2	sylan 580	1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑀) → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 → 𝜓)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2109  ∀wral 3045  Tr wtr 5216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-v 3452  df-ss 3933  df-uni 4874  df-tr 5217

This theorem is referenced by:  ralabsobidv  44955