Theorem rexabsod 46225

Description: Deduction form of rexabso 46223. (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 46223	. 2 ⊢ ((Tr 𝑀 ∧ 𝐴 ∈ 𝑀) → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 ∧ 𝜓)))
3	1, 2	sylan 592	1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑀) → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 ∧ 𝜓)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2147  ∃wrex 3090  Tr wtr 5218

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-v 3456  df-ss 3901  df-uni 4873  df-tr 5219

This theorem is used by:  rexabsobidv  46227