Theorem bj-elisset 32056

Description: Remove from elisset 3188 dependency on ax-ext 2590 (and on df-cleq 2603 and df-v 3175). This proof uses only df-clab 2597 and df-clel 2606 on top of first-order logic. It only requires ax-1--7 and sp 2041. Use bj-elissetv 32055 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 32055	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑦 𝑦 = 𝐴)
2		bj-denotes 32052	. 2 ⊢ (∃𝑦 𝑦 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴)
3	1, 2	sylib 207	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)

Syntax hints:   → wi 4   = wceq 1475  ∃wex 1695   ∈ wcel 1977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-sb 1868  df-clab 2597  df-clel 2606

This theorem is referenced by:  bj-isseti  32058  bj-ceqsalt  32069  bj-ceqsalg  32072  bj-spcimdv  32078  bj-vtoclg1f  32103