Theorem bj-sbft 34957

Description: Version of sbft 2262 using Ⅎ', proved from core axioms. (Contributed by BJ, 19-Nov-2023.)

Assertion

Ref	Expression
bj-sbft	⊢ (Ⅎ'𝑥𝜑 → ([𝑡 / 𝑥]𝜑 ↔ 𝜑))

Proof of Theorem bj-sbft

Step	Hyp	Ref	Expression
1		spsbe 2085	. . 3 ⊢ ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑)
2		bj-nnfe 34913	. . 3 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → 𝜑))
3	1, 2	syl5 34	. 2 ⊢ (Ⅎ'𝑥𝜑 → ([𝑡 / 𝑥]𝜑 → 𝜑))
4		bj-nnfa 34910	. . 3 ⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
5		stdpc4 2071	. . 3 ⊢ (∀𝑥𝜑 → [𝑡 / 𝑥]𝜑)
6	4, 5	syl6 35	. 2 ⊢ (Ⅎ'𝑥𝜑 → (𝜑 → [𝑡 / 𝑥]𝜑))
7	3, 6	impbid 211	1 ⊢ (Ⅎ'𝑥𝜑 → ([𝑡 / 𝑥]𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537  ∃wex 1782  [wsb 2067  Ⅎ'wnnf 34905

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-sb 2068  df-bj-nnf 34906

This theorem is referenced by: (None)