Theorem bj-sbcex 37383

Description: Proof of sbcex 3752 when taking bj-df-sb 37382 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 1901	. 2 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ ∀𝑥(𝑥 = 𝑦 → 𝜑)) → ∃𝑦 𝑦 = 𝐴)
2		bj-df-sb 37382	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝐴 ∧ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
3		isset 3467	. 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 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2145  Vcvv 3453  [wsbc 3742

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2215  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-sbc 3743

This theorem is used by: (None)