Theorem bj-issettru 35747

Description: Weak version of isset 3487 without ax-ext 2703. (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 35746	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑧 𝑧 = 𝐴)
2		bj-denoteslem 35745	. 2 ⊢ (∃𝑧 𝑧 = 𝐴 ↔ 𝐴 ∈ {𝑦 ∣ ⊤})
3	1, 2	bitri 274	1 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ 𝐴 ∈ {𝑦 ∣ ⊤})

Syntax hints:   ↔ wb 205   = wceq 1541  ⊤wtru 1542  ∃wex 1781   ∈ wcel 2106  {cab 2709

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 1913  ax-6 1971  ax-7 2011  ax-8 2108

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-clel 2810

This theorem is referenced by: (None)