Theorem sepgi 5259

Description: Inference associated with sepg 5258. The requirement that 𝑦 not occur in 𝜑 is necessary, as notsep 5334 shows. (Contributed by NM, 21-Jun-1993.) (Revised by BJ, 14-Jul-2026.)

Hypothesis

Ref	Expression
sepgi.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
sepgi	⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))

Distinct variable groups:   𝑥,𝐴,𝑦   𝜑,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem sepgi

Step	Hyp	Ref	Expression
1		sepgi.1	. 2 ⊢ 𝐴 ∈ V
2		sepg 5258	. 2 ⊢ (𝐴 ∈ V → ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
3	1, 2	ax-mp 5	1 ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))

Syntax hints:   ↔ wb 209   ∧ wa 400  ∀wal 1566  ∃wex 1807   ∈ wcel 2141  Vcvv 3453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1571  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836

This theorem is referenced by:  inex1  5285