Theorem bj-sbcex 37071

Description: Proof of sbcex 3749 when taking bj-df-sb 37070 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 1882	. 2 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ ∀𝑥(𝑥 = 𝑦 → 𝜑)) → ∃𝑦 𝑦 = 𝐴)
2		bj-df-sb 37070	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝐴 ∧ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
3		isset 3462	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴)
4	1, 2, 3	3imtr4i 294	1 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1552   = wceq 1554  ∃wex 1793   ∈ wcel 2136  Vcvv 3448  [wsbc 3739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-12 2206  ax-ext 2728

This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1557  df-ex 1794  df-nf 1798  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-v 3450  df-sbc 3740

This theorem is referenced by: (None)