Theorem bj-issettru 35057

Description: Weak version of isset 3445 without ax-ext 2709. (Contributed by BJ, 24-Apr-2024.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-issettru	⊢ (∃𝑥 𝑥 = 𝐴 ↔ 𝐴 ∈ {𝑦 ∣ ⊤})

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem bj-issettru

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bj-denotes 35056	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑧 𝑧 = 𝐴)
2		bj-denoteslem 35055	. 2 ⊢ (∃𝑧 𝑧 = 𝐴 ↔ 𝐴 ∈ {𝑦 ∣ ⊤})
3	1, 2	bitri 274	1 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ 𝐴 ∈ {𝑦 ∣ ⊤})

Syntax hints:   ↔ wb 205   = wceq 1539  ⊤wtru 1540  ∃wex 1782   ∈ wcel 2106  {cab 2715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-clel 2816

This theorem is referenced by: (None)