Theorem iseqsetv-clel 2814

Description: Alternate proof of iseqsetv-cleq 2799. The expression ∃𝑥𝑥 = 𝐴 does not depend on a particular choice of the set variable. Use this theorem in contexts where df-cleq 2727 or ax-ext 2707 is not referenced elsewhere in your proof. It is proven from a specific implementation (class builder, axiom df-clab 2714) 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 2813	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ 𝐴 ∈ {𝑧 ∣ ⊤})
2		issettru 2813	. 2 ⊢ (∃𝑦 𝑦 = 𝐴 ↔ 𝐴 ∈ {𝑧 ∣ ⊤})
3	1, 2	bitr4i 278	1 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴)

Syntax hints:   ↔ wb 206   = wceq 1542  ⊤wtru 1543  ∃wex 1781   ∈ wcel 2114  {cab 2713

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2714  df-clel 2810

This theorem is referenced by:  elisset  2817  axprglem  5367  bj-issettruALTV  37168  bj-issetwt  37170  bj-vtoclg1f1  37212