Theorem bj-elisset 34590

Description: Remove from elisset 2834 dependency on ax-ext 2730 (and on df-cleq 2751 and df-v 3412). This proof uses only df-clab 2737 and df-clel 2831 on top of first-order logic. It only requires ax-1--7 and sp 2181. Use bj-elissetv 34589 instead when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-elisset	⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem bj-elisset

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bj-elissetv 34589	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑦 𝑦 = 𝐴)
2		bj-denotes 34584	. 2 ⊢ (∃𝑦 𝑦 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴)
3	1, 2	sylib 221	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)

Syntax hints:   → wi 4   = wceq 1539  ∃wex 1782   ∈ wcel 2112

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 1912  ax-6 1971  ax-7 2016  ax-8 2114

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1542  df-ex 1783  df-sb 2071  df-clab 2737  df-clel 2831

This theorem is referenced by:  bj-isseti  34592  bj-ceqsalt  34600  bj-ceqsalg  34603  bj-spcimdv  34609  bj-vtoclg1f  34632