Theorem iseqsetvlem 2823

Description: Lemma for iseqsetv-cleq 2824. (Contributed by Wolf Lammen, 17-Aug-2025.) (Proof modification is discouraged.)

Assertion

Ref	Expression
iseqsetvlem	⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑧 𝑧 = 𝐴)

Distinct variable group:   𝑥,𝑧,𝐴

Proof of Theorem iseqsetvlem

Step	Hyp	Ref	Expression
1		eqeq1 2764	. 2 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝐴 ↔ 𝑧 = 𝐴))
2	1	cbvexvw 2069	1 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑧 𝑧 = 𝐴)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1569  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-cleq 2752

This theorem is used by:  iseqsetv-cleq  2824