Theorem iseqsetv-cleq 2827

Description: Alternate proof of iseqsetv-clel 2843. The expression ∃𝑥𝑥 = 𝐴 does not depend on a particular choice of the set variable. The proof here avoids df-clab 2742, df-clel 2839 and ax-8 2149, but instead is based on ax-9 2157, ax-ext 2735 and df-cleq 2755. 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 2742, or you need df-cleq 2755 anyway. See the alternative version , not using df-cleq 2755 or ax-ext 2735 or ax-9 2157. (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 2826	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑧 𝑧 = 𝐴)
2		iseqsetvlem 2826	. 2 ⊢ (∃𝑦 𝑦 = 𝐴 ↔ ∃𝑧 𝑧 = 𝐴)
3	1, 2	bitr4i 281	1 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570  ∃wex 1812

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-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2755

This theorem is used by:  clelab  2908