Theorem iseqsetv-cleq 2816

Description: Alternate proof of iseqsetv-clel 2831. The expression ∃𝑥𝑥 = 𝐴 does not depend on a particular choice of the set variable. The proof here avoids df-clab 2731, df-clel 2827 and ax-8 2134, but instead is based on ax-9 2142, ax-ext 2724 and df-cleq 2744. In particular it still accepts 𝑥 ∈ 𝐴 being a primitive syntax term, not assuming any specific semantics (like elementhood in some form).

Use it in contexts where you want to avoid df-clab 2731, or you need df-cleq 2744 anyway. See the alternative version , not using df-cleq 2744 or ax-ext 2724 or ax-9 2142. (Contributed by Wolf Lammen, 6-Aug-2025.) (Proof modification is discouraged.)

Assertion

Ref	Expression
iseqsetv-cleq	⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴)

Distinct variable groups:   𝑥,𝐴   𝑦,𝐴

Proof of Theorem iseqsetv-cleq

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iseqsetvlem 2815	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑧 𝑧 = 𝐴)
2		iseqsetvlem 2815	. 2 ⊢ (∃𝑦 𝑦 = 𝐴 ↔ ∃𝑧 𝑧 = 𝐴)
3	1, 2	bitr4i 280	1 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴)

Syntax hints:   ↔ wb 208   = wceq 1550  ∃wex 1789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-9 2142  ax-ext 2724

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1790  df-cleq 2744

This theorem is referenced by:  clelab  2896