Theorem bj-sbcex 37301

Description: Proof of sbcex 3753 when taking bj-df-sb 37300 as definition. (Contributed by BJ, 19-Feb-2026.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-sbcex	⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)

Proof of Theorem bj-sbcex

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		exsimpl 1897	. 2 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ ∀𝑥(𝑥 = 𝑦 → 𝜑)) → ∃𝑦 𝑦 = 𝐴)
2		bj-df-sb 37300	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝐴 ∧ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
3		isset 3468	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴)
4	1, 2, 3	3imtr4i 295	1 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400  ∀wal 1567   = wceq 1569  ∃wex 1808   ∈ wcel 2142  Vcvv 3454  [wsbc 3743

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-12 2212  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-nf 1813  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-sbc 3744

This theorem is used by: (None)