Theorem bj-elisset 34195

Description: Remove from elisset 3505 dependency on ax-ext 2793 (and on df-cleq 2814 and df-v 3496). This proof uses only df-clab 2800 and df-clel 2893 on top of first-order logic. It only requires ax-1--7 and sp 2182. Use bj-elissetv 34194 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 34194	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑦 𝑦 = 𝐴)
2		bj-denotes 34189	. 2 ⊢ (∃𝑦 𝑦 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴)
3	1, 2	sylib 220	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)

Syntax hints:   → wi 4   = wceq 1537  ∃wex 1780   ∈ wcel 2114

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 1970  ax-7 2015  ax-8 2116

This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1540  df-ex 1781  df-sb 2070  df-clab 2800  df-clel 2893

This theorem is referenced by:  bj-isseti  34197  bj-ceqsalt  34205  bj-ceqsalg  34208  bj-spcimdv  34214  bj-vtoclg1f  34237