Theorem bj-elissetALT 34963

Description: Alternate proof of elisset 2821. This is essentially the same proof as seen by inlining bj-denotes 34958 and bj-denoteslem 34957. Use elissetv 2820 instead when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem bj-elissetALT

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elissetv 2820	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑦 𝑦 = 𝐴)
2		bj-denotes 34958	. 2 ⊢ (∃𝑦 𝑦 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴)
3	1, 2	sylib 221	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)

Syntax hints:   → wi 4   = wceq 1543  ∃wex 1787   ∈ wcel 2112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2073  df-clab 2717  df-clel 2818

This theorem is referenced by: (None)