Theorem iseqsetv-clel 2810

Description: Alternate proof of iseqsetv-cleq 2795. The expression ∃𝑥𝑥 = 𝐴 does not depend on a particular choice of the set variable. Use this theorem in contexts where df-cleq 2723 or ax-ext 2703 is not referenced elsewhere in your proof. It is proven from a specific implementation (class builder, axiom df-clab 2710) 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 2809	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ 𝐴 ∈ {𝑧 ∣ ⊤})
2		issettru 2809	. 2 ⊢ (∃𝑦 𝑦 = 𝐴 ↔ 𝐴 ∈ {𝑧 ∣ ⊤})
3	1, 2	bitr4i 278	1 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴)

Syntax hints:   ↔ wb 206   = wceq 1541  ⊤wtru 1542  ∃wex 1780   ∈ wcel 2111  {cab 2709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-clel 2806

This theorem is referenced by:  elisset  2813  bj-issettruALTV  36924  bj-issetwt  36926  bj-vtoclg1f1  36968