Theorem bj-issettruALTV 37474

Description: Moved to main as issettru 2839 and kept for the comments.

Weak version of isset 3467 without ax-ext 2733. (Contributed by BJ, 24-Apr-2024.) (Proof modification is discouraged.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem bj-issettruALTV

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iseqsetv-clel 2840	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑧 𝑧 = 𝐴)
2		issettru 2839	. 2 ⊢ (∃𝑧 𝑧 = 𝐴 ↔ 𝐴 ∈ {𝑦 ∣ ⊤})
3	1, 2	bitri 278	1 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ 𝐴 ∈ {𝑦 ∣ ⊤})

Syntax hints:   ↔ wb 209   = wceq 1568  ⊤wtru 1569  ∃wex 1807   ∈ wcel 2141  {cab 2739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1571  df-ex 1808  df-sb 2095  df-clab 2740  df-clel 2836

This theorem is referenced by: (None)