Theorem rexabsod 45415

Description: Deduction form of rexabso 45413. (Contributed by Eric Schmidt, 19-Oct-2025.)

Hypothesis

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

Assertion

Ref	Expression
rexabsod	⊢ ((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 rexabsod

Step	Hyp	Ref	Expression
1		ralabsod.1	. 2 ⊢ (𝜑 → Tr 𝑀)
2		rexabso 45413	. 2 ⊢ ((Tr 𝑀 ∧ 𝐴 ∈ 𝑀) → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 ∧ 𝜓)))
3	1, 2	sylan 586	1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑀) → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 ∧ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∈ wcel 2119  ∃wrex 3063  Tr wtr 5179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-v 3433  df-ss 3900  df-uni 4839  df-tr 5180

This theorem is referenced by:  rexabsobidv  45417