Theorem iseqsetv-clel 2841

Description: Alternate proof of iseqsetv-cleq 2826. The expression ∃𝑥𝑥 = 𝐴 does not depend on a particular choice of the set variable. Use this theorem in contexts where df-cleq 2754 or ax-ext 2734 is not referenced elsewhere in your proof. It is proven from a specific implementation (class builder, axiom df-clab 2741) of the primitive term 𝑥 ∈ 𝐴. (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑦,𝐴

Proof of Theorem iseqsetv-clel

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		issettru 2840	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ 𝐴 ∈ {𝑧 ∣ ⊤})
2		issettru 2840	. 2 ⊢ (∃𝑦 𝑦 = 𝐴 ↔ 𝐴 ∈ {𝑧 ∣ ⊤})
3	1, 2	bitr4i 281	1 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1569  ⊤wtru 1570  ∃wex 1808   ∈ wcel 2142  {cab 2740

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-clel 2837

This theorem is used by:  elisset  2844  axprglem  5406  bj-issettruALTV  37536  bj-issetwt  37538  bj-vtoclg1f1  37580